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Рассел Б. - Математическая логика основанная на теории типов .pdf

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KjKönig, ‘Über die Grundlagen der Mengenlehre und das Kontinuum-problem’, Math. Annalen, Vol. LXI
(1905); A.C. Dixon, ‘On “well-ordered” aggregates’, Proc. London Math. Soc., Series 2, Vol. IV, Part I
(1906); E.W. Hobson, ‘On the Arithmetic Continuum’, ibidJ_r_gb_ij_^eh`_ggh_\ihke_^g_cbawlbo
klZl_cg_dZ`_lkyfg_Z^_d\Zlguf
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Kj.: Poincaré, ‘Les mathématiques et la logique’, Revue de Métaphysique et de Morale (May, 1906), hkh[_g
gh jZa^_eu VII b IX; kf., lZd`_, Peano, Revista de Mathematica, Vol. VIII, No.5 (1906), K. 149 b ^Ze__.
4
‘Una questione sui numeri transfiniti’, Rendiconti del circolo matematico di Palermo, Vol. XI (1897).
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eufqbkehflhnbf__l\k_k\hckl\Zij_^iheZ]Z_fu_bqbkeZfbke_^mxsbfbaZ\k_
fb l_fb qbkeZfb dhlhju_ ij_^iheZ]Zxl wlb k\hckl\Z¶ Gh a^_kv njZaZ µ\k_ k\hckl\Z¶
^he`gZ[ulvaZf_g_gZg_dhlhjhc^jm]hcnjZahcdhlhjZyaZdjulZ^eyl_o`_kZfuo\ha
jZ`_gbcFh`gh^himklblvqlhnjZaZµ\k_k\hckl\Zij_^iheZ]Z_fu_bqbkeZfbke_
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1
2
Kj. fhx klZlvx: ‘Les paradoxes de la logique’, Revue de Métaphysique et de Morale (May, 1906), K. 645.
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fgbfhc i_j_f_gghc fu g_ fh`_f hkms_kl\blv dZdhc-eb[h \u\h^ b ihwlhfm \h \k_o
^hdZaZl_evkl\Zo ^he`gu bkihevah\Zlvky ^_ckl\bl_evgu_ i_j_f_ggu_ Ij_^iheh`bf
\havfzfijhkl_crbc kemqZc gZf ba\_klgh qlhµφo \k_]^Z bklbggh¶ l_µx).φx¶ b fu
agZ_fqlhµφo\k_]^Z\e_qzlψo¶l_µx).{φx\e_qzlψx`¶DZdbfh[jZahffu\u\_^_fµψo
\k_]^Zbklbggh¶"FuagZ_fqlh\k_]^Zbklbgghke_^mxs___keb φo –bklbgghb_keb φo
\e_qzl ψolh ψo –bklbgghGhmgZkg_lihkuehd\lhfkfuke_qlh φo –bklbgghb φo
\e_qzlψomgZk_klvke_^mxs__ φo \k_]^Zbklbgghb φo \k_]^Z\e_qzlψo>eylh]hqlh
[uhkms_kl\blvgZr\u\h^fu^he`gui_j_clbhlµφo\k_]^Zbklbggh¶dφobhlµφo\k_
]^Z\e_qzl ψo¶dµφo\e_qzl ψo¶]^_wlhlohklZ\ZykvdZdbf-lh\hafh`gufZj]mf_glhf
^he`_g[ulvh^bgZdh\uf\h[hbokemqZyoLh]^Zbaµφx¶ bµφo\e_qzl ψo¶fu\u\h^bf
‘ψx¶lZdbfh[jZahf ψoy\ey_lkybklbgguf^eyex[h]h\hafh`gh]hZj]mf_glZbke_^h
\Zl_evgh bklbgguf \k_]^Z KlZeh [ulv ^ey lh]h qlh[u \u\_klb µx).ψx¶ ba µx).φx¶ b
Wlbfb^\mfyl_jfbgZfbfuh[yaZguI_Zghdhlhjucbkihevam_lboijb[ebabl_evgh\mdZaZgghf\u
r_kfuke_KjgZijbf_jFormulaire Mathématique (Turin, 1903), Vol. IV, C. 5.
2
F-jFZdDhee]h\hjblqlhµijhihabpbb¶^_eylkygZljbdeZkkZ^hklh\_jgu_i_j_f_ggu_bg_\ha
fh`gu_Fufh`_fijbgylvwlh^_e_gb_\ijbf_g_gbbdijhihabpbhgZevgufnmgdpbyfNmgdpby
dhlhjmxfh`ghml\_j`^Zlvy\ey_lky^hklh\_jghcnmgdpbydhlhjmxfh`ghhljbpZlvy\ey_lkyg_
\hafh`ghc\k_^jm]b_nmgdpbby\eyxlky\kfuke_f-jZFZdDheeZi_j_f_ggufb
3
Kf. _]h Grundgesetze der Arithmetik (Jena, 1893), lhf I, § 17, C. 31.
1
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gbyoh^ghcbeb[he__ijhihabpbhgZevguonmgdpbcdml\_j`^_gbxh\k_oagZq_gbyog_
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bf_xsb_ jZ\gu_ m]eu ijb hkgh\Zgbb y\eyxlky jZ\gh[_^j_ggufb¶ < qZklghklb wlhl
ijhp_kk lj_[m_lky ijb ^hdZaZl_evkl\_ Barbara b ^jm]bo fh^mkh\ kbeeh]bafZ >jm]bfb
keh\Zfb\kydZy^_^mdpbyhi_jbjm_l^_ckl\bl_evgufbi_j_f_ggufbbebdhgklZglZfb
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h]jZgbqb\rbkvdZdhc-lh\dZq_kl\_aZf_gu^ey\k_Wlhh^gZdhg_bf__lf_klZ<hav
fzfgZijbf_jhij_^_e_gb_g_ij_ju\ghcnmgdpbbijhpblbjh\Zggh_\ur_<wlhfhi
j_^_e_gbbσ, εbebδ^he`gu[ulvfgbfufbi_j_f_ggufbFgbfu_i_j_f_ggu_ihklh
yggh lj_[mxlky \ hij_^_e_gbyo <havfzf gZijbf_j lZdh_ µP_eh_ qbkeh gZau\Z_lky
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\hafh`guoagZq_gbcn’.
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qZxsbfnjZaZf>jm]bfbkeh\Zfbh[hagZqZxsZynjZaZhij_^_ey_lkyihkj_^kl\hfijh
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qlh[uwlbijhihabpbbijbh[j_lZebk\hzagZq_gb_q_j_ah[hagZqZxsb_njZaufu^he`
gu gZclb g_aZ\bkbfmx bgl_jij_lZpbx ijhihabpbc kh^_j`Zsbo lZdb_ njZau b g_
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1
2
Logic, qZklv I, jZa^_e II.
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y\ey_lkyagZqbfuflZdbfh[jZahf_kebaZ^ZlvdZdmx-lhhij_^_ezggmxnmgdpbx φolh
kms_kl\m_lijhihabpby]h\hjysZyhµ\k_owe_f_glZo\uihegyxsbo φo¶Ghbgh]^ZdZd
fum\b^bfiha`_kemqZ_lkylZdqlhlhqlh\_j[Zevghijhy\ey_lkydZdh^gZnmgdpby
gZkZfhf^_e_ij_^klZ\ey_lkh[hcfgh]hZgZeh]bqguonmgdpbckjZaebqgufbh[eZklyfb
agZqbfhklbWlhgZijbf_jijbf_gbfhdµj –bklbggh¶dhlhjZydZdfugZc^zfgZkZ
fhf^_e__klvg_h^gZnmgdpbyhljghij_^klZ\ey_lkh[hcjZaebqgu_nmgdpbbkhhl\_l
kl\mxsb_\b^mijhihabpbbdhlhjhcy\ey_lkyj<lZdhfkemqZ_njZaZ\ujZ`ZxsZyg_
hij_^_ezggmx nmgdpbx fh`_l [eZ]h^Zjy wlhc g_hij_^_ezgghklb [ulv agZqbfhc \h
\kzffgh`_kl\_agZq_gbcZj]mf_glZij_\hkoh^ysbfh[eZklvagZqbfhklbdZdhc-lhh^ghc
nmgdpbb<wlhfkemqZ_\k_g_h[hkgh\ZgghKlZeh[ulv_kebfuiulZ_fkykdZaZlvµ<k_
bklbggu_ijhihabpbbh[eZ^Zxlk\hckl\hf φ¶l_µ³j –bklbggh´\k_]^Z\e_qzl φj¶\ha
1
Ebg]\bklbq_kdb m^h[gh_ \ujZ`_gb_ ^ey wlhc b^_b ke_^mxs__ µφo – bklbggh ^ey \k_o \hafh`guo
agZq_gbco¶\hafh`gh_agZq_gb_ihgbfZ_lkydZdagZq_gb_^eydhlhjh]hφoy\ey_lkyagZqbfhc
fh`gu_ Zj]mf_glu ^ey ³j – bklbggh´ g_h[oh^bfh ij_\urZxl \hafh`gu_ Zj]mf_glu
^ey φbke_^h\Zl_evghjZkkfZljb\Z_fh_h[s__\ukdZau\Zgb_g_\hafh`ghIhwlhcijb
qbg_ih^ebgguoh[sbo\ukdZau\Zgbch\k_obklbgguoijhihabpbyok^_eZlvg_evayH^
gZdh fh`_l kemqblvky qlh ij_^iheZ]Z_fZy nmgdpby φ ih^h[gh ³j – bklbggh´ y\ey_lky
g_hij_^_ezgghcb_kebkemqblkyqlhhgZh[eZ^Z_lg_hij_^_ezgghklvxlhqghlZdh]h`_
\b^ZdZdb³j –bklbggh´fu\k_]^Z[m^_f\khklhygbbaZ^Zlvbgl_jij_lZpbx^eyijh
ihabpbbµ³j –bklbggh´\e_qzl φj¶Wlhijhbahc^zlgZijbf_j_keb φj_klvµg_-j –eh`
gh¶LZdbfh[jZahf\wlbokemqZyofuihemqZ_f\b^bfhklvh[sboijhihabpbcjZkkfZl
jb\Zxsbo \k_ ijhihabpbb gh wlZ \b^bfhklv k\hbf ihy\e_gb_f h[yaZgZ kbkl_fZlbq_
kdhcg_hij_^_ezgghklblZdbokeh\dZdbklbgghbeh`ghWlZkbkl_fZlbq_kdZyg_hij_
^_ezgghklv \ul_dZ_l ba b_jZjobb ijhihabpbc dhlhjZy [m^_l h[tykg_gZ iha^g__ <h
\k_olZdbokemqZyofufh`_f\ukdZaZlvkyhdZdhc-lhijhihabpbbihkdhevdmagZq_gb_
g_hij_^_ezgguokeh\[m^_lijbkihkZ[eb\ZlvkydwlhcdZdhc-lhijhihabpbbGh_kebfu
ij_h[jZam_f gZrm ijhihabpbx k ihfhsvx fgbfuo i_j_f_gguo b g_qlh kdZ`_f h[h
\kzf fu ^he`gu ij_^iheZ]Zlv g_hij_^_ezgghklv keh\ aZnbdkbjh\Zggmx \ lhf beb
bghf\hafh`ghfkfuke_ohlyfh`_l[ulvkh\_jr_ggh[_ajZaebqghlhdZdbfbak\hbo
\hafh`guo kfukeh\ hgb ^he`gu h[eZ^Zlv <hl lZd b kemqZ_lky qlh \ukdZau\Zgby h[h
\k_o bf_xl h]jZgbq_gby dhlhju_ bkdexqZxl µ\k_ ijhihabpbb¶ b l_f g_ f_g__ h^gh
\j_f_gghdZ`mlkybklbggufb\ukdZau\Zgbyfbh[hµ\k_oijhihabpbyo¶H[ZwlboimgdlZ
klZgmlykg__dh]^Z[m^_lh[tykg_gZl_hjbylbih\
QZklh ij_^iheZ]Zehkv 1 qlh ^ey lh]h qlh[u h[hkgh\Zggh ]h\hjblv h \k_o we_f_glZo
kh\hdmighklblj_[m_lkyqlh[ukh\hdmighklv[ueZdhg_qghcLZdµ<k_ex^bkf_jlgu¶
h[hkgh\Zgghihkdhevdmex^bh[jZamxldhg_qgucdeZkkGhgZkZfhf^_e_wlhg_ijb
qbgZihdhlhjhcfufh`_f]h\hjblvhµ\k_oex^yo¶DZd\b^ghbaijb\_^zggh]h\ur_
h[km`^_gby kms_kl\_ggZ g_ dhg_qghklv gh lhqlh fh`gh [ueh[u gZa\Zlv eh]bq_kdhc
h^ghjh^ghklvxWlhk\hckl\h^he`ghijbgZ^e_`Zlvex[hckh\hdmighklbqvbwe_f_glu
kmlv\k_we_f_glukh^_j`Zsb_ky\jZfdZoh[eZklbagZqbfhklbg_dhlhjhch^ghcnmgd
pbb ?keb ^_eh g_ \ kdjulhc g_hij_^_ezgghklb h[sbo eh]bq_kdbo l_jfbgh\ lZdbo dZd
bklbgghbeh`ghdhlhjZyijb^Zzl\b^bfhklv_^bghcnmgdpbblhfmqlhgZkZfhf^_e_
y\ey_lkydhg]ehf_jZlhffgh]bonmgdpbckjZaebqgufbh[eZklyfbagZqbfhklblhki_j
\h]h\a]ey^Z\k_]^Z\b^ghij_^iheZ]Z_lkh\hdmighklvwlhk\hckl\hbeb`_g_l
<u\h^u wlh]h jZa^_eZ aZdexqZxlky \ ke_^mxs_f DZ`^Zy ijhihabpby kh^_j`ZsZy
\k_ml\_j`^Z_lqlhg_dhlhjZyijhihabpbhgZevgZynmgdpby\k_]^ZbklbggZbwlhih^jZ
amf_\Z_lqlh\k_agZq_gbymdZaZgghcnmgdpbby\eyxlkybklbggufbghwlhg_ih^jZam
f_\Z_lqlhnmgdpbyy\ey_lkybklbgghc^ey\k_oZj]mf_glh\ihkdhevdm_klvZj]mf_glu
^ey dhlhjuo dZdZy-lh ^ZggZy nmgdpby y\ey_lky [_kkfuke_gghc l_ g_ bf__l agZq_gby
Ke_^h\Zl_evghfufh`_f]h\hjblvh\k_owe_f_glZokh\hdmighklblh]^Zblhevdhlh]^Z
dh]^Z kh\hdmighklv h[jZam_l qZklv beb p_eh_ h[eZklb agZqbfhklb g_dhlhjhc ijhihab
pbhgZevghc nmgdpbb ]^_ h[eZklv agZqbfhklb hij_^_ey_lky dZd kh\hdmighklv l_o Zj]m
f_glh\ ^ey dhlhjuo jZkkfZljb\Z_fZy nmgdpby y\ey_lky agZqbfhc l_ bf__l agZq_gb_
[value].
IVB?J:JOBYLBIH<
Lbihij_^_ey_lkydZdh[eZklvagZqbfhklbijhihabpbhgZevghcnmgdpbbl_dZdkh\h
dmighklv Zj]mf_glh\ ^ey dhlhjuo mdZaZggZy nmgdpby bf__l agZq_gby <k_]^Z dh]^Z \
1
GZijbf_j, ImZgdZj_. Kf.: Revue de Métaphysique et de Morale (May, 1906).
ijhihabpbb \klj_qZ_lky fgbfZy i_j_f_ggZy h[eZklv agZq_gbc fgbfhc i_j_f_gghc y\
ey_lky lbihf lbi nbdkbjm_lky nmgdpb_c hlghkbl_evgh dhlhjhc jZkkfZljb\Zxlky µ\k_
agZq_gby¶ G_h[oh^bfhklv jZaebq_gby h[t_dlh\ gZ lbiu \ua\ZgZ j_ne_dkb\gufb g_^h
jZamf_gbyfbdhlhju_\hagbdZxl_keblZdh]hjZaebqbyg_ijh\_klbDZdfu\b^_ebwlb
g_^hjZamf_gby ke_^m_l ba[_]Zlv k ihfhsvx lh]h qlh fh`_l [ulv gZa\Zgh µijbgpbihf
ihjhqgh]hdjm]Z¶l_µp_ehklghklvg_fh`_lkh^_j`Zlvwe_f_gluhij_^_ezggu_\l_j
fbgZo_zkZfhc¶<gZr_fl_ogbq_kdhfyaud_wlhlijbgpbiklZgh\blkyke_^mxsbfµLh
qlh kh^_j`bl fgbfmx i_j_f_ggmx g_ ^he`gh [ulv \hafh`guf agZq_gb_f wlhc i_j_
f_gghc¶ LZdbf h[jZahf \kz qlh kh^_j`bl fgbfmx i_j_f_ggmx ^he`gh hlghkblky d
lbimhlebqghfmhl\hafh`guoagZq_gbcwlhci_j_f_gghcfu[m^_f]h\hjblvqlhhgh
hlghkblkyd[he__\ukhdhfmlbimLZdbfh[jZahffgbfu_i_j_f_ggu_kh^_j`Zsb_ky\
\ujZ`_gbbkmlvlhqlhhij_^_ey_l_]hlbiWlh–\_^msbcijbgpbi\^Zevg_cr_fba
eh`_gbb
Ijhihabpbb dhlhju_ kh^_j`Zl fgbfu_ i_j_f_ggu_ \hagbdZxl ba ijhihabpbc g_
kh^_j`Zsbowlbofgbfuoi_j_f_gguoihkj_^kl\hfijhp_kkh\h^bgbadhlhjuo\k_]^Z
y\ey_lky ijhp_kkhf h[h[s_gby l_ ih^klZgh\dhc i_j_f_gghc \f_klh h^gh]h ba l_jfb
gh\ ijhihabpbb b ml\_j`^_gb_f j_amevlbjmxs_c nmgdpbb ^ey \k_o \hafh`guo agZq_
gbcwlhci_j_f_gghcKe_^h\Zl_evghijhihabpbygZau\Z_lkyh[h[szgghc,dh]^ZhgZkh
^_j`bl fgbfmx i_j_f_ggmx Ijhihabpbx g_ kh^_j`Zsmx fgbfuo i_j_f_gguo fu
[m^_fgZau\Zlvwe_f_glZjghcijhihabpb_cYkghqlhijhihabpbykh^_j`ZsZyfgbfu_
i_j_f_ggu_ij_^iheZ]Z_l^jm]b_ijhihabpbbbadhlhjuohgZfh`_l[ulvihemq_gZih
kj_^kl\hf h[h[s_gby ke_^h\Zl_evgh \k_ h[h[szggu_ ijhihabpbb ij_^iheZ]Zxl we_
f_glZjgu_ijhihabpbb<we_f_glZjghcijhihabpbbfufh`_fjZaebqblvh^bgbeb[h
e__qe_gh\hlh^gh]hbeb[he__ihgylbc; qe_gukmlvlhqlhfh`_ljZkkfZljb\ZlvkydZd
km[t_dl ijhihabpbb lh]^Z dZd ihgylby y\eyxlky ij_^bdZlZfb beb hlghr_gbyfb ml
\_j`^Z_fufbhlghkbl_evghwlbol_jfbgh\1Qe_guwe_f_glZjguoijhihabpbcfu[m^_f
gZau\Zlvbg^b\b^Zfbhgbh[jZamxli_j\ucbebgbarbclbi
GZijZdlbd_g_h[yaZl_evghagZlvdZdb_h[t_dluijbgZ^e_`Zlgbar_fmlbimg_h[y
aZl_evgh^Z`_agZlvy\ey_lkyebgbarbclbii_j_f_gguo\klj_qZxsboky\^Zgghfdhg
l_dkl_ lbihf bg^b\b^h\ beb `_ dZdbf-lh ^jm]bf B[h gZ ijZdlbd_ bf_xl agZq_gb_
lhevdhhlghkbl_evgu_ lbiu i_j_f_gguo ihwlhfm gbarbc lbi \klj_qZxsbcky \ ^Zg
ghfdhgl_dkl_fh`_l[ulvgZa\Zglbihfbg^b\b^h\ihklhevdmihkdhevdmjZkkfZljb\Z
_lkywlhldhgl_dklHlkx^Zke_^m_lqlhijb\_^zggh_\ur_jZkkfhlj_gb_bg^b\b^h\g_
kms_kl\_ggh^eybklbgghklblh]hqlhb^zl^Ze__kms_kl\_g_glhevdhkihkh[dhlhjuf
ba bg^b\b^h\ ijhba\h^ylky ^jm]b_ lbiu L_f g_ f_g__ lbi bg^b\b^h\ fh`gh h[jZah
\Zlv
Ijbf_gyy ijhp_kkh[h[s_gby dbg^b\b^Zf \oh^ysbf \ we_f_glZjgu_ ijhihabpbb
fu ihemqZ_f gh\u_ ijhihabpbb H[hkgh\Zgghklv wlh]h ijhp_kkZ lj_[m_l lhevdh lh]h
qlh[ubg^b\b^ug_[uebijhihabpbyfbLhqlhwlhlZd^he`ghh[_ki_qb\Zlvkykfuk
ehfdhlhjucfuijb^Zzfkeh\mbg^b\b^Fufh`_fhij_^_eblvbg^b\b^dZdg_qlheb
rzggh_dhfie_dkghklblh]^Zhq_\b^ghqlhhgg_y\ey_lkyijhihabpb_cihkdhevdmijh
ihabpbbkms_kl\_gghdhfie_dkguKe_^h\Zl_evgh\ijbf_g_gbbijhp_kkZh[h[s_gbyd
bg^b\b^Zffug_ih^\_j`_gujbkdm\iZklv\j_ne_dkb\gu_g_^hjZamf_gby
We_f_glZjgu_ijhihabpbb\kh\hdmighklbkl_fbijhihabpbyfbdhlhju_\dZq_kl\_
fgbfuoi_j_f_gguokh^_j`Zllhevdhbg^b\b^ufu[m^_fgZau\Zlvijhihabpbyfbi_j
\h]hihjy^dZHgbh[jZamxl\lhjhceh]bq_kdbclbi
1
KfPrinciples of Mathematics, § 48.
LZdbf h[jZahf fu bf__f gh\mx p_ehklghklv p_ehklghklv ijhihabpbc i_j\h]h ih
jy^dZKlZeh[ulvfufh`_fh[jZah\Zlvgh\u_ijhihabpbb\dhlhju_i_j\hihjy^dh\u_
ijhihabpbb\oh^yldZdfgbfu_i_j_f_ggu_Bofu[m^_fgZau\Zlvijhihabpbyfb\lh
jh]h ihjy^dZ hgb h[jZamxl lj_lbc eh]bq_kdbc lbi LZd gZijbf_j _keb Wibf_gb^ ml
\_j`^Z_lµ<k_ijhihabpbbi_j\h]hihjy^dZml\_j`^Z_fu_fghceh`gu¶hgml\_j`^Z_l
ijhihabpbx \lhjh]h ihjy^dZ hg ^_ckl\bl_evgh fh`_l _x ml\_j`^Zlv g_ ml\_j`^Zy \
^_ckl\bl_evghklb dZdhc-lh i_j\hihjy^dh\hc ijhihabpbb b ihwlhfm ijhlb\hj_qby g_
\hagbdZ_l
MdZaZgguc \ur_ ijhp_kk fh`gh ijh^he`Zlv [_kdhg_qgh n + 1-uc eh]bq_kdbc lbi
[m^_lkhklhylvbaijhihabpbcihjy^dZndhlhju_[m^ml\dexqZlvijhihabpbbihjy^dZn
–ghg_[he__\ukhdh]hihjy^dZq_fihjy^hdfgbfuoi_j_f_gguoIhemq_ggu_lZdbf
h[jZahf lbiu \aZbfhbkdexqZxsb b ihwlhfm j_ne_dkb\gu_ g_^hjZamf_gby g_\hafh`
gu^hl_oihjihdZfuihfgbfqlhfgbfu_i_j_f_ggu_^he`gu\k_]^Zh]jZgbqb\Zlvky
jZfdZfbg_dhlhjh]hh^gh]hlbiZ
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jZaebqguoihjy^dh\fh]ml[ulvihemq_gubaijhihabpbcjZaebqguoihjy^dh\f_lh^hf
ih^klZgh\db?kebj –ijhihabpbybZ –dhgklblm_glZjlhimklvµj/Z;o¶hagZqZ_lijhih
abpbxdhlhjZyihemqZ_lkyijbih^klZgh\d_o\f_klhZ\_a^_]^_Z\oh^bl\jLh]^Zj/Z,
dhlhjmxfu[m^_fgZau\ZlvfZljbp_cfh`_laZgylvf_klhnmgdpbb_zagZq_gb_^eyZj
]mf_glZo_klvj/Z;oZ_zagZq_gb_^eyZj]mf_glZ Z_klvjKoh^gufh[jZahf_kebµj/(Z,
b);(o, m¶hagZqZ_lj_amevlZli_j\hcih^klZgh\dbo\f_klhZZaZl_fih^klZgh\dbm\f_klh
b fu fh`_f bkihevah\Zlv ^\mof_klgmx fZljbpm j/(a, b ^ey lh]h qlh[u ij_^klZ\blv
^\mof_klgmx nmgdpbx Wlbf kihkh[hf fu fh`_f ba[_`Zlv fgbfuo i_j_f_gguo hl
ebqguohlbg^b\b^h\bijhihabpbcjZaebqguoihjy^dh\Ihjy^hdfZljbpu[m^_lhij_
^_eylvkydZdihjy^hdijhihabpbb\dhlhjhcijhba\_^_gZih^klZgh\dZZkZfmwlmijhih
abpbx fu [m^_f gZau\Zlv ijhlhlbihf Ihjy^hd fZljbpu g_ hij_^_ey_l _z lbi \hi_j\uoihlhfmqlhhgZg_hij_^_ey_lqbkehZj]mf_glh\\f_klhdhlhjuo^he`gu[ulv
ih^klZ\e_gu^jm]b_Zj]mf_glul_bf__lebfZljbpZnhjfmj/Z, j/(Z, bbebj/(Z, b, k)’
l^ \h-\lhjuo ihlhfm qlh _keb ijhlhlbi hlghkblky d [he__ \ukhdhfm q_f i_j\uc
ihjy^dmZj]mf_glufh]ml[ulveb[hijhihabpbyfbeb[hbg^b\b^ZfbGhykghqlhlbi
fZljbpu\k_]^Zhij_^_ebfihkj_^kl\hfb_jZjobbijhihabpbc
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aZf_gblv ijhlhlbi j gZ φZ b aZf_gblv j/Z;o gZ φo lZdbf h[jZahf lZf ]^_ dZd fgbfu_
i_j_f_ggu_ihy\eyebkv[ujbZ_keb[uijbf_gyeZkvfZljbpZ\dZq_kl\_gZr_cfgb
fhc i_j_f_gghc fu l_i_jv bf__f φ. >ey hijZ\^Zgby φ \ dZq_kl\_ fgbfhc i_j_f_gghc
g_h[oh^bfhqlh[u_zagZq_gbyh]jZgbqb\Zebkvijhihabpbyfbg_dhlhjh]hh^gh]hlbiZ
Ihwlhfmfuijh^he`Z_fke_^mxsbfh[jZahf
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qZxsZyi_j\hihjy^dh\mxnmgdpbxbebijhihabpbx\dZq_kl\_fgbfhci_j_f_gghc[m
^_l gZau\Zlvky \lhjhihjy^dh\hc nmgdpb_c b l^ Nmgdpby hl h^ghc i_j_f_gghc hlgh
kysZykydihjy^dmke_^mxs_fmaZihjy^dhf_zZj]mf_glZ[m^_lgZau\Zlvkyij_^bdZlb\
ghc nmgdpb_c lZdh_ `_ gZa\Zgb_ [m^_l ^Z\Zlvky nmgdpbb hl g_kdhevdbo i_j_f_gguo
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ij_^bdZlb\ghc dh]^Z agZq_gby ijbibku\Zxlky \k_f ^jm]bf i_j_f_gguf Lh]^Z lbi
nmgdpbbhij_^_ey_lkylbihf_zagZq_gbcbqbkehfblbihf_zZj]mf_glh\
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\h]h ihjy^dZ hl bg^b\b^Z o [m^_l h[hagZqZlvky dZd φ!o ^ey nmgdpbc [m^ml lZd`_ bk
ihevah\Zlvky[md\u ψ, χ, θ, f, g, F, GG_i_j\hihjy^dh\u_nmgdpbbkh^_j`Zlnmgdpbx
\dZq_kl\_fgbfhci_j_f_gghcke_^h\Zl_evghlZdb_nmgdpbbh[jZamxl\iheg_hij_^_
ezggmxp_ehklghklvb φ\ φ!ofh`_l[ulvij_h[jZah\ZgZ\fgbfmxi_j_f_ggmxEx[Zy
ijhihabpby\dhlhjhc φihy\ey_lkydZdfgbfZyi_j_f_ggZyb\dhlhjhcg_lfgbfuoi_
j_f_gguo[he__\ukhdh]hq_f φlbiZy\ey_lkyijhihabpb_c\lhjh]hihjy^dZ?keblZ
dZyijhihabpbykh^_j`blbg^b\b^ohgZg_y\ey_lkyij_^bdZlb\ghcnmgdpb_chlogh
_kebhgZkh^_j`bli_j\hihjy^dh\mxnmgdpbx φhgZy\ey_lkyij_^bdZlb\ghcnmgdpb_c
hl φb[m^_laZibku\ZlvkydZdf!(ψ! ∧z Lh]^Zf_klvij_^bdZlb\gZynmgdpby\lhjh]hih
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< ij_^u^ms_c klZlv_ ^ey wlh]h `mjgZeZ \ dZq_kl\_ g_hij_^_ey_fhc \f_klh ^batxgdpbb y [jZe bf
iebdZpbx <u[hj f_`^m gbfb – wlh ij_^f_l \dmkZ L_i_jv y \u[bjZx ^batxgdpbx ihkdhevdm hgZ
iha\hey_lgZffbgbfbabjh\Zlvqbkehbkoh^guoijhihabpbc[Kf.: ‘The Theory of Implication’, American Journal of Mathematics, Vol. XXVIII, 1906, P. 159-202.]
qbkehlhq_d1<o).φo ogZau\Z_lkyfgbfhci_j_f_gghcdh]^Z φoml\_j`^Z_lkylZf]^_
og_mlhqgzgogZau\Z_lky^_ckl\bl_evghci_j_f_gghc.
DZdZy-lh ij_^bdZlb\gZy nmgdpby hl Zj]mf_glZ dZdh]h-lh lbiZ ih h[klhyl_evkl
\Zf hgZ [m^_l ij_^klZ\e_gZ dZd φ!o, φ!α beb φ!R Ij_^bdZlb\gZy nmgdpby hl o – wlh
nmgdpby qvb agZq_gby y\eyxlky ijhihabpbyfb hlghkysbfbky d lbim ke_^mxs_fm aZ
lbihf o _keb o y\ey_lkybg^b\b^hfbeb ijhihabpb_c beb aZ lbihf agZq_gbc o _keb o
y\ey_lkynmgdpb_cHgZfh`_l[ulvhibkZgZdZdnmgdpby\dhlhjhc\k_fgbfu_i_j_
f_ggu__keblZdh\u__klvhlghkylkydh^ghfmlbimkobebdf_gvr_fmlbimI_j_f_g
gZyhlghkblkydf_gvr_fmq_folbim_kebhgZfh`_lagZqbfh\klj_qZlvkydZdZj]mf_gl
\kZfhf obebdZdZj]mf_gl\Zj]mf_gl_kZfh]hobl^
Ml\_j`^_gb_l_ml\_j`^_gb_qlhg_dhlhjZyijhihabpbyy\ey_lkybklbgghcbeb
qlh dZdh_-lh agZq_gb_ g_dhlhjhc ijhihabpbhgZevghc nmgdpbb y\ey_lky bklbgguf Ml
\_j`^_gb_ lj_[m_lky ^ey lh]h qlh[u hlebqblv ^_ckl\bl_evgh ml\_j`^Z_fmx ijhihab
pbxhlijhihabpbbijhklhjZkkfZljb\Z_fhcbebhlijhihabpbbgZdhlhjmxkkueZxlky
dZdgZmkeh\b_g_dhlhjhc^jm]hcijhihabpbbGZml\_j`^_gb_[m^_lmdZau\ZlvagZdµL¶
ij_^ihkeZgguc lhfm qlh ml\_j`^Z_lky k ^hklZlhqguf dhebq_kl\hf lhq_d qlh[u aZ
dexqblvlhqlhml\_j`^Z_lky\kdh[db2.
I_j_^ l_f dZd i_j_clb d bkoh^guf ijhihabpbyf gZf gm`gu g_dhlhju_ hij_^_e_
gby<ke_^mxsbohij_^_e_gbyolZd`_dZdb\bkoh^guoijhihabpbyo[md\up, q, rbk
ihevamxlky^eyh[hagZq_gbyijhihabpbc
p ⊃ q . = . ∼p ∨ q Df.
Wlhhij_^_e_gb_mklZgZ\eb\Z_lqlhµp ⊃ q¶dhlhjh_ijhqblu\Z_lkydZdµj\e_qzlq’)
^he`ghhagZqZlvµj –eh`ghbebq –bklbggh¶Yg_gZf_j_\Zxkvml\_j`^Zlvqlhµ\e_
qzl¶g_fh`_lbf_lv^jm]h]hkfukeZghml\_j`^ZxlhevdhlhqlhwlhlkfukegZb[he__
ih^oh^bl ^ey lh]h qlh[u aZ^Zlv µ\e_qzl¶ \ kbf\hebq_kdhc eh]bd_ < hij_^_e_gbb agZd
jZ\_gkl\Z b [md\u µDf¶ ^he`gu jZkkfZljb\Zlvky dZd h^bg kbf\he kh\f_klgh hagZqZy
µagZqblihhij_^_e_gbx¶AgZdjZ\_gkl\Z[_a[md\µDf¶bf__lbghckfukedhlhjuc\kdh
j_[m^_ljZkkfhlj_g
p . q . = . ∼(∼p ∨ ∼q)
Df.
Wlhhij_^_ey_leh]bq_kdh_ijhba\_^_gb_^\moijhihabpbcjbql_µjbqh[Zy\ey
xlky bklbggufb¶ Ijb\_^zggh_ hij_^_e_gb_ mklZgZ\eb\Z_l qlh wlh ^he`gh hagZqZlv
µEh`ghqlhj –eh`ghbebq –eh`gh¶A^_kvhij_^_e_gb_kgh\Zg_^Zzl_^bgkl\_ggh]h
kfukeZdhlhjucfh`_l [ulvijb^Zgµjbqh[Zy\eyxlkybklbggufb¶ gh aZ^Zzl agZq_
gb_dhlhjh_gZb[he__ih^oh^bl^eygZr_cp_eb
p ≡ q . = . p ⊃ q . q ⊃ p Df.
Ijbbkihevah\Zgbblhq_dfuke_^m_fI_ZghWlh bkihevah\Zgb_ iheghklvx h[tykg_gh f-jhf MZclo_
^hf; kf.: ‘On Cardinal Numbers’, American Journal of Mathematics, Vol, XXIV, b ‘On Mathematical Concepts of Material World’, Phil. Trans. A., Vol. CCV, P. 472.
2
WlbfagZdhfdZdb\\_^_gb_fb^_bdhlhjmxhg\ujZ`Z_lfuh[yaZguNj_]_Kf_]hBegriffsschrift
(Halle, 1879), C>Jmkkdbci_j_\h^kfBkqbke_gb_ihgylbcNj_]_=Eh]bdZbeh]bq_kdZyk_fZglb
dZ–F:ki_dlIj_kk@bGrundgesetze der Arithmetik (Jena, 1983), Vol. I, C. 9.
1
Lh _klv µp ≡ q¶ dhlhjh_ qblZ_lky dZd µj wd\b\Ze_glgh q¶ hagZqZ_l µj \e_qzl q b q
\e_qzlj¶hldm^Zdhg_qghke_^m_lqlhjbqy\eyxlkyh[Zbklbggufbbebh[Zeh`gufb
(∃
∃o) . φo . = . ∼{(x) . ∼φx}
Df.
Wlhhij_^_ey_lµKms_kl\m_lihdjZcg_cf_j_h^ghagZq_gb_o^eydhlhjh]h φoy\ey
_lkybklbgguf¶Fuhij_^_ey_fihke_^g__dZdhagZqZxs__µEh`ghqlh φo\k_]^Zeh`
gh¶
x = y . = : (φ) : φ!x . ⊃ . φ!y
Df.
Wlh–hij_^_e_gb_jZ\_gkl\ZHghmklZgZ\eb\Z_lqlhobm^he`gugZau\ZlvkyjZ\
gufb dh]^Z dZ`^Zy ij_^bdZlb\gZy nmgdpby \uihegyxsZyky o \uihegy_lky m Ba Zd
kbhfuk\h^bfhklbke_^m_lqlh_kebo\uihegy_l ψo]^_ ψ_klvdZdZy-lhnmgdpbyij_
^bdZlb\gZybebg_ij_^bdZlb\gZylhm\uihegy_lψm.
Ke_^mxsb_hij_^_e_gbyf_g__\Z`gub\\h^ylkylhevdhkp_evxkhdjZs_gby
(x, y) . φ(x, y) . = : (x) : (y) . φ(x, y)
Df,
Df,
(∃
∃x, y) . φ(x, y) . = : (∃
∃x) : (∃
∃y) . φ(x, y)
φx . ⊃x . ψx : = : (x) : φx ⊃ ψx Df,
φx . ≡x . ψx : = : (x) : φx . ≡ . ψx Df,
φ(x, y) . ⊃x, y . ψ(x, y) : = : (x, y) : φ(x, y) . ⊃ . ψ(x, y)
Df,
bl^^eyex[h]hqbkeZi_j_f_gguo
Lj_[mxlky ke_^mxsb_ bkoh^gu_ ijhihabpbb \ b p, q, r h[hagZqZxl
ijhihabpbb
(1) Ijhihabpby\u\_^_ggZybabklbgghcihkuedby\ey_lkybklbgghc
(2) L: p ∨ p . ⊃ . p.
(3) L: q . ⊃ . p ∨ q.
(4) L: p ∨ q . ⊃ . q ∨ p.
(5) L: p ∨ (q ∨ r) . ⊃ . q ∨ (p ∨ r).
(6) L: . q ⊃ r . ⊃ : p ∨ q . ⊃ . p ∨ r.
(7) L: (x) . φx . ⊃ . φy;
l_µ_keb\k_agZq_gbyφ ∧o y\eyxlkybklbggufblhφmy\ey_lkybklbgguf]^_φm_klvdZ
dh_-lhagZq_gb_¶1.
?keb φm – bklbggh ]^_ φm _klv dZdh_-lh agZq_gb_ φ ∧o lho).φo – bklbggh Wlh]h
g_evay\ujZablv\gZrbokbf\heZob[h_kebfuaZibku\Z_fµφm . ⊃ . (o)φo¶wlhhagZqZ_l
‘φm\e_qzlqlh\k_agZq_gby φ ∧o y\eyxlkybklbggufb]^_mfh`_lijbgbfZlvex[h_agZ
q_gb_ih^oh^ys_]hlbiZ¶qlh\h[s_fg_bf__lf_klZLhqlhfugZf_j_\Z_fkyml\_j
`^ZlvaZdexqZ_lky\ke_^mxs_fµ?kebijbex[hf\u[jZgghfm φm –bklbgghlho).φo –
bklbggh¶lh]^ZdZdlhqlh\ujZ`_ghihkj_^kl\hfµφy . ⊃ . (x) . φx¶_klvµIjbex[hf\u
1
M^h[ghbkihevah\ZlvaZibkvφoqlh[uh[hagZqblvkZfmnmgdpbx\ijhlb\hiheh`ghklvlhfmbebbgh
fmagZq_gbxwlhcnmgdpbb
[jZgghfm_kebφm –bklbgghlho).φo –bklbggh¶qlhy\ey_lkykh\_jr_gghbguf\ukdZ
au\Zgb_fdhlhjh_\h[s_fkemqZ_eh`gh
(9) L: (o) . φo . ⊃ . φZ]^_Z_klvdZdZy-lhhij_^_ezggZydhgklZglZ
WlhijbgpbigZkZfhf^_e_ij_^klZ\ey_lkh[hcfgh]hjZaebqguoijbgpbih\Zbf_g
gh klhevdh kdhevdh kms_kl\m_l \hafh`guo agZq_gbc Z L_ hg mklZgZ\eb\Z_l gZijb
f_j qlh lh qlh bf__l kbem ^ey \k_o bg^b\b^h\ bf__l kbem ^ey KhdjZlZ Z lZd`_ hgh
bf__l kbem ^ey IeZlhgZb l^ Wlhl ijbgpbi khklhbl \ lhf qlhh[s__ijZ\beh fh`gh
ijbf_gblvdqZklghfmkemqZxghqlh[uaZ^Zlv_]hh[eZklvg_h[oh^bfhmihfygmlvhl
^_evgu_ijbf_juihkdhevdm\ijhlb\ghfkemqZ_gZfgm`_gijbgpbidhlhjuckZfaZ\_
jylgZk\h[s_fijZ\be_qlhh[sb_ijZ\beZdhlhju_fh]mlijbf_g_gudqZklghfmkem
qZx fh]ml [ulv ijbf_g_gu d hl^_evghfm kemqZx kdZ`_f d KhdjZlm LZdbf h[jZahf
wlhlijbgpbihlebqZ_lkyhl^Zggucijbgpbi\ukdZau\Z_lkyhKhdjZl_IeZlhg_beb
dZdhc-lh^jm]hcdhgklZgl_lh]^ZdZd\ukdZau\Z_lkyhi_j_f_gghc
MdZaZgguc ijbgpbi gbdh]^Z g_ bkihevam_lky \ kbf\hebq_kdhc eh]bd_ beb \ qbklhc
fZl_fZlbd_ihkdhevdm\k_gZrbijhihabpbby\eyxlkyh[sbfbB^Z`_lh]^Zdh]^ZdZd
\µh^bg_klvqbkeh¶fuih\b^bfhklbbf__fkljh]hqZklguckemqZcijb[ebadhfjZk
kfhlj_gbbhgg_hdZau\Z_lkylZdh\ufNZdlbq_kdbijbf_g_gb_wlh]hijbgpbiZy\ey_l
ky hlebqbl_evguf ijbagZdhf ijbdeZ^ghc fZl_fZlbdb KlZeh [ulv kljh]h ]h\hjy fu
^he`gubkdexqblv_]hbagZr_]hkibkdZ
(10) L: . (o) . j ∨ φo . ⊃ : j . ∨ . (o) . φo;
l_µ_keb³jbebφo” –\k_]^Zbklbgghlhbebj –bklbgghbebφo –\k_]^Zbklbggh¶
Dh]^Zf(φx) –bklbgghijbex[hf\hafh`ghfZj]mf_gl_obF(φy) –bklbgghijb
ex[hf\hafh`ghfZj]mf_gl_mlh]^Z^f(φx) . F(φx`y\ey_lkybklbggufijbex[hf\ha
fh`ghfZj]mf_gl_o.
Wlh–ZdkbhfZµg_hij_^_ezgghklbi_j_f_gguo¶HgZgm`gZdh]^ZhdZ`^hcba^\mo
hl^_evguoijhihabpbhgZevguonmgdpbcba\_klghqlhhgb\k_]^Zy\eyxlkybklbggufb
b fu ohlbf \u\_klb qlh bo eh]bq_kdh_ ijhba\_^_gb_ \k_]^Z y\ey_lky bklbgguf Wlhl
\u\h^ hijZ\^Zg lhevdh lh]^Z dh]^Z ^\_ nmgdpbb ijbgbfZxl Zj]mf_glu h^gh]h b lh]h
`_lbiZb[h\ijhlb\ghfkemqZ_boeh]bq_kdh_ijhba\_^_gb_[_kkfuke_ggh
?keb φo.φo⊃ψo –bklbggh^eyex[h]h\hafh`gh]holh ψo –bklbggh^eyex[h]h
\hafh`gh]ho.
WlZ ZdkbhfZ lj_[m_lky ^ey lh]h qlh[u aZ\_jblv gZk \ lhf qlh h[eZklv agZqbfhklb
ψo\ij_^iheZ]Z_fhfkemqZ_kh\iZ^Z_lkh[eZklvxagZqbfhklbφo.φo⊃ψo.⊃.ψonZdlbq_
kdbh[_h[eZklbkh\iZ^Zxlkh[eZklvxagZqbfhklbφo<ij_^iheZ]Z_fhfkemqZ_fuagZ
_f qlh ψo – bklbggh \_a^_ ]^_ b φo.φo⊃ψo b φo.φo⊃ψo.⊃.ψo y\eyxlky agZqbfufb gh
[_aZdkbhfufug_agZ_fqlh ψo –bklbggh\_a^_]^_ ψoy\ey_lkyagZqbfufKe_^h\Z
l_evghwlZZdkbhfZgZfg_h[oh^bfZ
:dkbhfublj_[mxlkygZijbf_jijb^hdZaZl_evkl\_
(o) . φo : (o) . φo ⊃ ψo : ⊃ . (o) . ψo.
Ihb
L: . (o) . φo : (o) . φo ⊃ ψo : ⊃ : φm . φm ⊃ ψm,
hlkx^Zih
L: . (o) . φo : (o) . φo ⊃ ψo : ⊃ : ψm,
hlkx^Zj_amevlZl\ul_dZ_lihb
(13) L: . (∃
∃f) : . (x) : φx . ≡ . f!x.
Wlh – ZdkbhfZ k\h^bfhklb HgZ mklZgZ\eb\Z_l qlh _keb aZ^Zlv dZdmx-lh nmgdpbx
∧
φ o lhkms_kl\m_llZdZyij_^bdZlb\gZynmgdpbyf! ∧o qlhf!x\k_]^Zwd\b\Ze_glgZ φoAZ
∃f¶ihhij_^_e_gbx_klvhljbpZgb_
f_lbfqlhihkdhevdmijhihabpbygZqbgZxsZykykµ∃
ijhihabpbbgZqbgZxs_ckykµf¶ijb\_^zggZyZdkbhfZ\dexqZ_l\hafh`ghklvjZkkfhl
j_gby µ\k_o ij_^bdZlb\guo nmgdpbc hl o¶ ?keb φo _klv dZdZy-lh nmgdpby hl o fu g_
∃φ¶ihkdhevdmfug_fh`_f
fh`_f\ukdZaZlvijhihabpbxgZqbgZxsmxkykµφ¶bebµ∃
jZkkfZljb\Zlv µ\k_ nmgdpbb¶ gh lhevdh µdZdmx-lh nmgdpbx¶ beb µ\k_ ij_^bdZlb\gu_
nmgdpbb¶
(14) L: . (∃
∃f) : . (x, y) : φ(x, y) . ≡ . f!(x, y).
Wlh–ZdkbhfZk\h^bfhklb^ey^\mof_klghcnmgdpbb
< ijb\_^zgguo \ur_ ijhihabpbyo gZrb o b m fh]ml hlghkblvky d ex[hfm lbim
?^bgkl\_ggh_ ]^_ mf_klgZ l_hjby lbih\ khklhbl \ lhf qlh ebrv iha\hey_l gZf
hlh`^_kl\blv ^_ckl\bl_evgu_ i_j_f_ggu_ \klj_qZxsb_ky \ jZaebqguo kh^_j`Zgbyo
dh]^Z^_fhgkljbjm_lkyqlhhgbhlghkylkydh^ghfmblhfm`_lbimihkdhevdm\h[hbo
kemqZyo\oh^yldZdZj]mf_gluh^ghcblhc`_nmgpbbbqlh\bmbZkhhl\_lkl
\_ggh^he`guhlghkblkydlbimih^oh^ys_fm^eyZj]mf_glh\ φ ∧z Ihwlhfm_kebij_^
iheh`blvgZijbf_jqlhmgZk_klvijhihabpbynhjfuφ).f!(φ! ∧z , xy\eyxsZyky\lhjh
ihjy^dh\hcnmgdpb_chlolhih
L: (φ) . f!(φ! ∧z , x) . ⊃ . f!(ψ! ∧z , x),
]^_ ψ! ∧z _klvdZdZy-lhnmgdpbyi_j\h]hihjy^dZGhφ) . f!(φ! ∧z , xg_evayjZkkfZljb\Zlv
lZddZd_keb[uhgZ[ueZi_j\hihjy^dh\hcnmgdpb_chlob[jZlvwlmnmgdpbxdZd\ha
fh`gh_agZq_gb_ ψ! ∧z \mdZaZgghf\ur_\ujZ`_gbbIh^h[gh_kf_r_gb_lbih\ijb\h
^bldiZjZ^hdkme`_pZ.
Kgh\Z jZkkfhljbf deZkku dhlhju_ g_ y\eyxlky qe_gZfb kZfbo k_[y Ykgh qlh ih
kdhevdmfuhlh`^_kl\ey_fdeZkkuknmgdpbyfb1gbh[h^ghfdeZkk_g_evayagZqbfh]h
\hjblvqlhhgy\ey_lkybebg_y\ey_lkyqe_ghfkZfh]hk_[yb[hqe_gudeZkkZy\eyxlky
Zj]mf_glZfb nmgdpbb Z Zj]mf_glu nmgdpbb \k_]^Z hlghkylky d lbim [he__ gbadhfm
q_fnmgdpbyB_kebfukijhkbfµDZdh[klhbl^_ehkdeZkkhf\k_odeZkkh\"Hgqlh`_
g_ y\ey_lky deZkkhf b ihwlhfm qe_ghf kZfh]h k_[y"¶ hl\_l ^\hckl\_g_g <h-i_j\uo
_kebµdeZkk\k_odeZkkh\¶hagZqZ_lµdeZkk\k_odeZkkh\ex[h]hlbiZ¶lhlZdh]hihgylby
g_l <h-\lhjuo _keb µdeZkk \k_o deZkkh\¶ hagZqZ_l µdeZkk \k_o deZkkh\ lbiZ t¶ lh wlhl
deZkkhlghkblkydlbimke_^mxs_fmaZtZihlhfmkgh\Zg_y\ey_lkyqe_ghfk_[ykZfh
]h
LZdbf h[jZahf ohly ijb\_^zggu_ \ur_ ijhihabpbb jZ\guf h[jZahf ijbf_gyxlky
dh\k_flbiZfhgbg_iha\heyxlgZf\u\_klbijhlb\hj_qbyIhwlhfm\ijhp_kk_dZdhceb[h ^_^mdpbb gbdh]^Z g_ gm`gh jZkkfZljb\Zlv Z[khexlguc lbi i_j_f_gghc g_h[oh
^bfhebrv\b^_lvqlhjZaebqgu_i_j_f_ggu_\klj_qZxsb_ky\h^ghcijhihabpbbhl
ghkylkydgZ^e_`Zsbfkhhl\_lkl\mxsbflbiZfWlhbkdexqZ_ll_nmgdpbbbadhlhjuo
[uehihemq_ghgZr_q_l\zjlh_ijhlb\hj_qb_Zbf_gghµHlghr_gb_Rbf__lkbemf_`
^mRbS¶B[hhlghr_gb_f_`^mRbSg_h[oh^bfhhlghkblkyd[he__\ukhdhfmlbimq_f
ex[h_bagbolZdqlhij_^iheZ]Z_fZynmgdpbyy\ey_lky[_kkfuke_gghc
1
Wlhhlh`^_kl\e_gb_ih^e_`blfh^bnbdZpbbdhlhjZy\kdhj_[m^_lh[tykg_gZ
VIIWE?F?GL:JG:YL?HJBYDE:KKH<BHLGHR?GBC
Ijhihabpbb\dhlhju_\oh^blnmgdpby φfh]mlihk\h_fmbklbgghklghfmagZq_gbx
aZ\bk_lvhlhkh[hcnmgdpbbφbeb`_hgbfh]mlaZ\bk_lvhlh[tzfZφl_hlZj]mf_glh\
dhlhju_\uihegyxl φNmgdpbbihke_^g_]hkhjlZfu[m^_fgZau\Zlvwdkl_gkbhgZevgu
fbLZdgZijbf_jµY\_jxqlh\k_ex^bkf_jlgu¶g_fh`_l[ulvwd\b\Ze_glghµY\_
jxqlh\k_[_kizju_^\mgh]b_kf_jlgu¶^Z`__kebex^bihh[tzfmkh\iZ^Zxlk^\mgh
]bfb [_kizjufb b[h y fh]m b g_ agZlv qlh ih h[tzfm hgb h^bgZdh\u Gh µ<k_ ex^b
kf_jlgu¶ ^he`gh [ulv wd\b\Ze_glgh µ<k_ [_kizju_ ^\mgh]b_ kf_jlgu¶ _keb ex^b ih
h[tzfmkh\iZ^Zxlk^\mgh]bfbb[_kizjufbLZdbfh[jZahfµ<k_ex^bkf_jlgu¶y\ey
_lky wdkl_gkbhgZevghc nmgdpb_c hl nmgdpbb µo – q_eh\_d¶ lh]^Z dZd µY \_jx qlh \k_
ex^b kf_jlgu¶ g_ y\ey_lky wdkl_gkbhgZevghc nmgdpb_c fu [m^_f gZau\Zlv nmgdpbx
bgl_gkbhgZevghcdh]^ZhgZg_y\ey_lkywdkl_gkbhgZevghcNmgdpbbhlnmgdpbckdhlh
jufb hkh[h bf__l^_eh fZl_fZlbdZ \k_ y\eyxlky wdkl_gkbhgZevgufb IjbagZd wdkl_g
kbhgZevghcnmgdpbbfhlnmgdpbbφ! ∧z khklhbl\ke_^mxs_f
φ!o . ≡o . ψ!o : ⊃φ, ψ : f(φ! ∧z ) . ≡ . f(ψ! ∧z ).
Banmgdpbbfhlnmgdpbb φ! ∧z fufh`_f\u\_klbkhhl\_lkl\mxsmxwdkl_gkbhgZev
gmxnmgdpbxke_^mxsbfh[jZahfImklv
f{ ∧z (ψz)} . = : (∃
∃φ) : φ!x . ≡x . ψx : f{φ! ∧z }
Df.
Nmgdpbyf{ ∧z (ψz`nZdlbq_kdb_klvnmgdpbyhl ψ ∧z ohlyhgZbg_kh\iZ^Z_lknmgdpb_c
f(ψ! ∧z ij_^iheZ]Zy qlh wlZ ihke_^gyy y\ey_lky agZqbfhc Gh ljZdlh\Zlv lZd f{ ∧z (ψz)}
l_ogbq_kdb m^h[gh ohly hgZb kh^_j`bl Zj]mf_gl ∧z (ψz dhlhjuc fugZau\Z_fµdeZkk
hij_^_ey_fucihkj_^kl\hfψ¶Fubf__f
L: . φx . ≡x . ψx : ⊃ : f{ ∧z (φz)} . ≡ . f{ ∧z (ψz)},
ke_^h\Zl_evgh ijbf_gyy hij_^_e_gb_ lh`^_kl\Z d nbdlb\guf h[t_dlZf ∧z (φz b ∧z (ψz),
^Zggh_\ur_fugZoh^bfqlh
L: . φx . ≡x . ψx : ⊃ . ∧z (φz) = ∧z (ψz).
Wlhml\_j`^_gb_ZlZd`__]hdhg\_jkbyqlhlZd`_fh`gh^hdZaZlvmdZau\Z_lhleb
qbl_evgh_ k\hckl\h deZkkh\ Ke_^h\Zl_evgh fu \iheg_ fh`_f ljZdlh\Zlv ∧z (φz dZd
deZkkhij_^_ey_fucihkj_^kl\hfφL_f`_kZfufkihkh[hffumklZgZ\eb\Z_f
f{ ∧x ∧y ψ(x, y)} . = : (∃
∃φ) : φ!(x, y) . ≡x, y . ψ(x, y) : f{φ!( ∧x , ∧y )}
Df.
A^_kvg_h[oh^bfhg_kdhevdhkeh\hlghkbl_evghjZaebqbyf_`^m φ!( ∧x , ∧y b φ!( ∧y , ∧x Fu
[m^_f ijbgbfZlv ke_^mxs__ kh]eZr_gb_ Dh]^Z nmgdpby \ ijhlb\hiheh`ghklv k\hbf
agZq_gbyf ij_^klZ\e_gZ \ nhjf_ \dexqZxs_c ∧x b ∧y beb dZdb_-lh ^jm]b_ ^\_ [md\u
ZenZ\blZagZq_gb_wlhcnmgdpbb^eyZj]mf_glh\Zbb^he`ghh[gZjm`b\Zlvkyih^klZ
gh\dhc Z \f_klh ∧x b b \f_klh ∧y l_ Zj]mf_gl mihfbgZxsbcky i_j\uf ^he`_g ih^
klZ\eylvky \f_klh [md\u dhlhjZy \klj_qZ_lky \ ZenZ\bl_ jZgvr_ Z Zj]mf_gl mihfb
gZxsbcky \lhjuf – \f_klh [md\u dhlhjZy \klj_qZ_lky iha^g__ B wlh \iheg_ m^h\e_
l\hjbl_evghijh\h^bljZaebqb_f_`^mφ!( ∧x , ∧y bφ!( ∧y , ∧x gZijbf_j
AgZq_gb_φ!( ∧x ,
AgZq_gb_φ!( ∧x ,
AgZq_gb_φ!( ∧y ,
AgZq_gb_φ!( ∧y ,
FumklZgZ\eb\Z_f
ke_^h\Zl_evgh
∧
y ^eyZj]mf_glh\Zbb_klvφ!(Z,
∧
y ^eyZj]mf_glh\bbZ_klvφ!(b,
∧
x ^eyZj]mf_glh\Zbb_klvφ!(b,
∧
x ^eyZj]mf_glh\bba_klvφ!(a,
o∈φ! ∧z . = . φ!o
b).
Z).
a).
b).
Df.,
L: . x∈ ∧z (ψz) . = : (∃
∃φ) : φ!y . ≡y . ψy : φ!x.
Dlhfm`_ihZdkbhf_k\h^bfhklbfubf__f
(∃
∃φ) : φ!y . ≡y . ψy,
ke_^h\Zl_evgh
L: x∈ ∧z (ψz) . ≡ . ψx.
Wlh bf__l kbem ijb ex[hf o Ij_^iheh`bf l_i_jv qlh fu ohlbf jZkkfhlj_lv
∧
∧ ∧
z (ψz)∈ φ f{ z (φ!z`Kh]eZkghbaeh`_gghfm\ur_fubf__f
hlkx^Z
L: . ∧z (ψz)∈ φ∧ f{ ∧z (φ!z)} . ≡ . f{ ∧z (ψz)} : ≡ : (∃
∃φ) : φ!y . ≡y . ψy : f(φ!z),
L: . ∧z (ψz) = ∧z (χz) . ⊃ : ∧z (ψz)∈x . ≡χ . ∧z (χz)∈x,
]^_oaZibku\Z_lky\f_klhex[h]h\ujZ`_gbynhjfu φ∧ f{ ∧z (φ!z)}.
FumklZgZ\eb\Z_f
Df.
∃φ} . α = ∧z (φ!z)}
cls = α∧ {(∃
A^_kv cls h[eZ^Z_l agZq_gb_f dhlhjh_aZ\bkblhl lbiZ fgbfhc i_j_f_gghc φ Ke_^h\Z
l_evgh ijhihabpby µcls ∈ cls¶ gZijbf_j y\eyxsZyky ke_^kl\b_f ijb\_^zggh]h \ur_
hij_^_e_gbylj_[m_lqlhµcls’^he`ghh[eZ^ZlvjZaebqgufagZq_gb_f\^\mof_klZo]^_
hgh \klj_qZ_lky Kbf\he µcls’ fh`_l bkihevah\Zlvky lhevdh lZf ]^_ g_h[oh^bfh agZlv
lbihgh[eZ^Z_lg_hij_^_ezgghklvxdhlhjZyijbkihkZ[eb\Z_lkydh[klhyl_evkl\Zf?k
eb fu \\h^bf dZd g_hij_^_ey_fmx nmgdpbx µIndiv!x’ hagZqZxsmx µx – bg^b\b^¶ fu
fh`_fmklZgh\blv
Kl =
∧
∃φ}
α {(∃
. α = ∧z (φ!z . Indiv!z)}
Df.
Lh]^ZKl –wlhhij_^_ezgguckbf\hehagZqZxsbcµdeZkkbg^b\b^h\¶
Fu[m^_fbkihevah\Zlvkljhqgu_[md\u]j_q_kdh]hZenZ\blZbgu_q_f ∈, φ, ψ, χ,
θqlh[uij_^klZ\eylvdeZkkuex[h]hlbiZl_h[hagZqZlvkbf\heunhjfu ∧z (φ!zbeb
∧
z (φz).
Kwlh]himgdlZl_hjbydeZkkh\\hfgh]hfjZa\b\Z_lkydZd\kbkl_f_I_Zgh ∧z (φzaZ
f_gy_lz%(φzLZd`_ymklZgZ\eb\Zx
α ⊂ β . = : x∈α . ⊃ . x∈β Df.,
Df.,
∃!α . = . (∃
∃x) . x∈α
∧
V = x (x = x)Df.,
Λ = ∧x {∼(x = x)} Df.,
∃, Λ, VdZdbkbf\heuclsb ∈g_hij_
]^_ΛdZdbmI_Zgh_klvgmev-deZkkKbf\heu∃
^_e_gubijbh[j_lZxlhij_^_ezggh_agZq_gb_dh]^ZjZkkfZljb\Z_fuclbimdZaZgbguf
kihkh[hf
Hlghr_gbyfuljZdlm_flhqghlZdbf`_kihkh[hfmklZgZ\eb\Zy
a{φ!( ∧x , ∧y )}b . = . φ!(a, b)
Df.
ihjy^hd ij_^hij_^_ezg ZenZ\blguf ihjy^dhf o b m b lbih]jZnkdbf ihjy^dhf Z b b);
hlkx^Z
L: . a{ ∧x ∧y ψ(x, y)}b . ≡ : (∃
∃φ) : ψ(x, y) . ≡x, y . φ!(x, y) : φ!(a, b),
hldm^ZihZdkbhf_k\h^bfhklb
L: a{ ∧x ∧y ψ(x, y)}b . ≡ . ψ(a, b).
Bkihevamyijhibkgu_[md\ueZlbgkdh]hZenZ\blZ\dZq_kl\_khdjZs_gby^eylZdbokbf
\heh\dZd ∧x ∧y ψ(x, yfugZoh^bfqlh
]^_
FumklZgZ\eb\Z_f
L: . R = S . ≡ : xRy . ≡x, y . xSy,
R = S . = : f!R . ⊃f . f!S
Rel =
∧
R {∃
∃φ)
.R=
∧ ∧
x y φ!(x,
Df.
y)} Df.
b gZoh^bf qlh \kz qlh ^hdZau\Z_lky ^ey deZkkh\ bf__l k\hc ZgZeh] ^ey ^\mof_klguo
hlghr_gbcKe_^myI_ZghfumklZgZ\eb\Z_f
α∩β = ∧x (x∈α . x∈β)
Df.,
hij_^_eyyijhba\_^_gb_bebh[smxqZklv^\modeZkkh\
α∪β = ∧x (x∈α . ∨ . x∈β) Df.,
hij_^_eyykmffm^\modeZkkh\b
– α = ∧x {∼(x∈α)} Df.,
hij_^_eyyhljbpZgb_deZkkZKoh^gufh[jZahf^eyhlghr_gbcfumklZgZ\eb\Z_f
•
R ∩S =
•
R∪S =
∧ ∧
x y (xRy
∧ ∧
x y (xRy
. xSy)
Df.,
. ∨ . xSy)
•
∧
− R = x ∧y {∼(xRy)}
Df.,
Df.
VIII. >?KDJBILB<GU?NMGDPBB
Nmgdpbb jZkkfhlj_ggu_ ^h kbo ihj aZ bkdexq_gb_f g_kdhevdbo hl^_evguo nmgd
pbclZdbodZdR ∩ S[uebijhihabpbhgZevgufbGhh[uqgu_nmgdpbbfZl_fZlbdblZ
db_dZdo2, sin x, log xg_y\eyxlkyijhihabpbhgZevgufbNmgdpbbwlh]h\b^Z\k_]^Zha
gZqZxlµwe_f_glbf_xsbclZdh_-lhblZdh_-lhhlghr_gb_do¶Ihwlhcijbqbg_hgbfh
]ml [ulv gZa\Zgu ^_kdjbilb\gufb [descriptive@ nmgdpbyfb ihkdhevdm hgb hibku\Zxl
[describe@ hij_^_ezgguc we_f_gl q_j_a _]h hlghr_gb_ d bo Zj]mf_glZf LZd µsin π/2’
hibku\Z_lqbkehh^gZdhijhihabpbb\dhlhjuo\klj_qZ_lky πg_hklZgmlkyl_fb`_
kZfufb_keb[u\gbo[uehih^klZ\e_ghWlhgZijbf_jh[gZjm`b\Z_lkybaijhihab
pbbµsin π ¶dhlhjZykh^_j`blagZqbfmxbgnhjfZpbxlh]^ZdZdµ ¶–ljb\bZevgh
>_kdjbilb\gu_ nmgdpbb bf_xl agZq_gb_ g_ kZfb ih k_[_ gh lhevdh dZd dhgklblm_glu
ijhihabpbcbwlh\hh[s_ijbf_gy_lkydnjZaZfnhjfuµwe_f_glbf_xsbclZdh_-lhb
lZdh_-lhk\hckl\h¶Ke_^h\Zl_evghbf_y^_ehklZdbfbnjZaZfbfu^he`guhij_^_eylv
dZdmx-lhijhihabpbx\dhlhjmxhgb\oh^ylZg_njZamkZfmihk_[_1LZdbfh[jZahf
fu ijboh^bf d ke_^mxs_fm hij_^_e_gbx \ dhlhjhf µ x)(φx)’ ^he`gh qblZlvky dZd
‘^Zgguc [the@we_f_glxdhlhjuc\uihegy_lφo’.
•
∃b) : φx . =x . x=b : ψb
ψ{( x)(φx)} . = : (∃
Df.
Wlh hij_^_e_gb_ mklZgZ\eb\Z_l qlh µwe_f_gl dhlhjuc \uihegy_l φ \uihegy_l ψ’
^he`gh hagZqZlv µKms_kl\m_l l_jfbg b lZdhc qlh φo – bklbggh lh]^Z b lhevdh lh]^Z
dh]^Zo_klvbb ψb –bklbggh¶LZdbfh[jZahf\k_ijhihabpbbh[µ^ZgghflZdhf-lhb
lZdhf-lh¶[m^mleh`gufb_keblZdh]h-lhblZdh]h-lhg_kms_kl\m_lbebbokms_kl\m_l
g_kdhevdh
H[s__hij_^_e_gb_^_kdjbilb\ghcnmgdpbby\ey_lkyke_^mxsbf
R‘y = (
x)(xRy) Df.;
l_µR‘y¶^he`ghhagZqZlvµwe_f_gldhlhjucbf__lhlghr_gb_Rdm¶?keb`_kms_kl\m
_lg_kdhevdhbebg_kms_kl\m_lgbh^gh]hwe_f_glZbf_xs_]hhlghr_gb_Rdmlh\k_
ijhihabpbbhR‘y[m^mleh`gufbFumklZgZ\eb\Z_f
E!(
x)(φx) . = : (∃
∃b) : φx . ≡x . x=b
Df.
A^_kvµE!( x)(φx¶fh`_lijhqblu\ZlvkyµKms_kl\m_llZdhcwe_f_gldZdodhlhjuc\u
ihegy_lφo¶bebµlhlodhlhjuc\uihegy_lφokms_kl\m_l¶Fubf__f
L: . E!R‘y . ≡ : (∃
∃b) : xRy . ≡x . x=b.
1
Kfmihfygmlmx\ur_klZlvxµOn Denoting¶]^_ijbqbguwlh]hij_^klZ\e_gu[he__ijhkljZggh
DZ\uqdZ\R‘yfh`_lijhqblu\ZlvkyLZd_kebR –hlghr_gb_hlpZdkugmlhµR‘y¶_klv
µhl_pm¶?kebR –hlghr_gb_kugZdhlpm\k_ijhihabpbbhR‘y[m^mleh`gufb_kebm
g_bf__lgbh^gh]hbeb[hevr_q_fh^gh]hkugZ
BakdZaZggh]h\ur_h[gZjm`b\Z_lkyqlh^_kdjbilb\gu_nmgdpbbihemqZxlkybahl
ghr_gbc Hij_^_ey_fu_ l_i_jv hlghr_gby ]eZ\guf h[jZahf \Z`gu ^ey jZkkfhlj_gby
^_kdjbilb\guonmgdpbcdhlhjufhgb^ZxlgZqZeh
∧
∧
Cnv = Q P {xQy . ≡x, y . yPx}
Df.
A^_kvCnv_klvkhdjZs_gb_^eyµdhg\_jkby¶Wlhhij_^_ey_lhlghr_gb_g_dh]hhlghr_
gby d k\h_c dhg\_jkbbgZijbf_jhlghr_gb_ hlghr_gby[hevr_ dhlghr_gbx f_gvr_,
hlghr_gby hlph\kl\Z d hlghr_gbx kugh\kl\Z hlghr_gb_ ij_^r_kl\_ggbdZ d hlghr_
gbxgZke_^gbdZbl^Fubf__f
L. Cnv‘P = ( Q){xQy . ≡x, y . yPx}.
>eykhdjZs_gbyaZibkbqlhqZklh[he__m^h[ghfumklZgZ\eb\Z_f
∪
P = Cnv‘P Df.
GZflj_[m_lky_szh^gZaZibkv^eydeZkkZl_jfbgh\bf_xsbohlghr_gb_RdmKwlhc
p_evxfumklZgZ\eb\Z_f
→
∧
R = α ∧y {α = ∧x (xRy)}
hlkx^Z
Df.,
L. R ‘y = ∧x (xRy).
→
Koh^gufh[jZahffumklZgZ\eb\Z_f
←
∧
R = β ∧x {β = ∧y (xRy)}
hlkx^Z
Df.,
L. R ‘x = ∧y (xRy).
←
>Ze__gZflj_[m_lkyh[eZklvRl_deZkkwe_f_glh\bf_xsbohlghr_gb_Rdq_fmeb[hdhg\_jkgZyh[eZklvRl_deZkkwe_f_glh\ddhlhjufqlh-eb[hbf__lhlghr_gb_
Rbihe_ Rij_^klZ\eyxs__kh[hckmffmh[eZklbRbdhg\_jkghch[eZklbRKwlhcp_
evx fu hij_^_ey_f hlghr_gby h[eZklb dhg\_jkghc h[eZklb b ihey d R Hij_^_e_gby
lZdh\u
∧
∧
∃y) . xRy)}
D = α R {α = ∧x ((∃
∧
Df.,
∧
∃x) . xRy)} Df.,
[D] = β R {β = ∧y ((∃
∧ ∧
∃y) : xRy . ∨ . yRx)}
C = γ R {γ = ((∃
Df.
AZf_lbf qlh lj_lv_ ba wlbo hij_^_e_gbc agZqbfh lhevdh lh]^Z dh]^Z R _klv lh qlh
fh`gh [ueh [u gZa\Zlv h^ghjh^guf hlghr_gb_ l_ hlghr_gb_f \ dhlhjhf _keb xRy
bf__lf_klhobmhlghkylkydh^ghfmblhfm`_lbim<ijhlb\ghfkemqZ_dZd[ufug_
\u[bjZebobmeb[hxRyeb[hyRx[ueb[u[_kkfuke_ggufbWlhgZ[ex^_gb_\Z`gh\
k\yabkiZjZ^hdkhf;mjZeb-Nhjlb
GZhkgh\Zgbbijb\_^zgguohij_^_e_gbcfuihemqZ_f
∃y) . xRy},
L. D‘R = ∧x {(∃
∧
∃x) . xRy},
L. [D]‘R = y {(∃
∃y) : xRy . ∨ . yRx},
L. C‘R = ∧x {(∃
ihke_^g__[m^_l agZqbfh lhevdh lh]^Z dh]^Z R h^ghjh^ghµD‘R¶ qblZ_lky dZdµh[eZklv
R’; ‘[D]‘R¶qblZ_lkydZdµdhg\_jkgZyh[eZklvR’; ‘C‘R¶qblZ_lkydZdµihe_R’.
>Ze__gZflj_[m_lkyaZibkv^eyhlghr_gbydeZkkZqe_gh\ddhlhjufg_dhlhjucwe_
f_gl ba α bf__l hlghr_gb_ R d deZkkm α kh^_j`Zs_fmky \ h[eZklb R Z lZd`_ aZibkv
^eyhlghr_gbydeZkkZqe_gh\dhlhju_bf_xlhlghr_gb_Rdg_dhlhjhfmwe_f_glmba β,
ddeZkkm βkh^_j`Zs_fmky\dhg\_jkghch[eZklbR>ey\lhjhcbagbofumklZgZ\eb\Z
_f
∧
Ihwlhfm
∧
∃y) . y∈β . xRy)}
R∈ = α β {α = ∧x ((∃
Df.
∃y) . y∈β . xRy}.
L. R∈‘β = ∧x {(∃
LZd_kebR_klvhlghr_gb_hlpZdkugmZ β –wlhdeZkk\uimkdgbdh\BlhgZlhR∈‘β[m
^_ldeZkkhfµhlpu\uimkdgbdh\BlhgZ¶_kebR_klvhlghr_gb_µf_gvr_¶Zβ –wlhdeZkk
ijZ\bevguo ^jh[_c nhjfu –2–n ^ey p_euo agZq_gbc n lh R∈‘β [m^_l deZkkhf ^jh[_c
f_gvrboq_fg_dhlhjZy^jh[vnhjfu–2–nl_R∈‘β[m^_ldeZkkhfijZ\bevguo^jh[_c
>jm]h_\ur_mihfygmlh_hlghr_gb__klv R )∈.
<dZq_kl\_Zevl_jgZlb\ghcaZibkbqZklh[he__m^h[ghcfumklZgZ\eb\Z_f
∪
R‘‘β = R∈‘β Df.
Hlghkbl_evgh_ ijhba\_^_gb_ ^\mo hlghr_gbc R b S _klv hlghr_gb_ dhlhjh_ bf__l
f_klhf_`^mobz\k_]^Zdh]^Zbf__lkywe_f_glmlZdhcqlhbxRybyRzbf_xlf_klh
Hlghkbl_evgh_ijhba\_^_gb_h[hagZqZ_lkydZdRSLZd
RS =
FulZd`_mklZgZ\eb\Z_f
∧∧
∃y)
x z {(∃
. xRy . yRz}
R2 = RR
Df.
Df.
QZklh lj_[mxlky ijhba\_^_gb_ b kmffZ deZkkZ deZkkh\ Hgb hij_^_eyxlky ke_^mx
sbfh[jZahf
∃α) . α∈κ . x∈α}
s‘κ = ∧x {(∃
∧
p‘κ = x {α∈κ . ⊃α . x∈α}
Df.
Df.
Koh^gufh[jZahf^eyhlghr_gbcfumklZgZ\eb\Z_f
.
s ‘λ = ∧x ∧y {(∃
∃R) . R∈λ . xRy}
Df.
.
p ‘λ = ∧x ∧y {R∈λ . ⊃R . xRy}
Df.
GZfgm`gZaZibkv^eydeZkkh\qvbf_^bgkl\_ggufwe_f_glhfy\ey_lkyoI_Zghbk
ihevam_lιxihwlhfmfu[m^_fbkihevah\Zlvι‘xI_ZghihdZaZewlhih^qzjdb\ZebNj_
]_ qlh wlhl deZkkg_evay hlh`^_kl\blv k o Ijbh[uqghf \a]ey^_gZ deZkkug_h[oh^b
fhklv lZdh]h jZaebqby hklZzlky aZ]Z^hqghc gh k lhqdb aj_gby \u^\bgmlhc \ur_ hgZ
klZgh\blkyhq_\b^ghc
FumklZgZ\eb\Z_f
∧
ι = α ∧x {α = ∧y (y = x)}
hlkx^Z
Df.,
L. ι‘x = ∧y (y = x) Df.,
b
L: E! ι ‘α . ⊃ . ι ‘α = (
∪
∪
x)(x∈α);
l__keb α –wlhdeZkkdhlhjucbf__llhevdhh^bgwe_f_gllhwlbfwe_f_glhfy\ey_lky
∪
ι ‘α .
1
>eydeZkkZdeZkkh\kh^_j`Zsboky\^ZgghfdeZkk_fumklZgZ\eb\Z_f
∧
Cl‘α = β (β ⊂ α) Df.
L_i_jvfufh`_fi_j_clbdjZkkfhlj_gbxdZj^bgZevguobhj^bgZevguoqbk_eblh
]hdZdboaZljZ]b\Z_lmq_gb_hlbiZo
IXD:J>BG:EVGU?QBKE:
DZj^bgZevgh_qbkehdeZkkZ αhij_^_ey_lkydZddeZkk\k_odeZkkh\koh^guok α^\Z
deZkkZ y\eyxlky koh^gufb dh]^Z f_`^m gbfb bf__lky h^gh-h^ghagZqgh_ hlghr_gb_
DeZkkh^gh-h^ghagZqguohlghr_gbch[hagZqZ_lkydZd→bhij_^_ey_lkyke_^mxsbf
h[jZahf
∧
1→1 = R {xRy . x/Ry . xRy/ . ⊃x, y, x/, y/ . x = x/ . y = y/}
Df.
Koh^kl\hh[hagZqZ_lkydZdSimbhij_^_ey_lkylZd
∧
∧
Sim = α β {(∃
∃R) . R∈1→1 . D‘R = α . D‘R = β}
Df.
Lh]^Z Sim ‘α_klvihhij_^_e_gbxdZj^bgZevgh_qbkeh α_]hfu[m^_fh[hagZqZlvdZd
Nc‘αke_^h\Zl_evghfumklZgZ\eb\Z_f
→
1
∪
LZdbfh[jZahf ι ‘α_klvlhqlhI_ZghgZau\Z_lια.
→
Nc = Sim
hlkx^Z
Df.,
L. Nc‘α = Sim ‘α.
→
DeZkkdZj^bgZevguoqbk_efu[m^_fh[hagZqZlvdZdNClZdbfh[jZahf
NC = Nc‘‘cls
Df.
hij_^_ey_lky dZd deZkk qvbf _^bgkl\_gguf we_f_glhf y\ey_lky gmev-deZkk l_ Λ),
ihwlhfm
0 = ι‘Λ
Df.
Hij_^_e_gb_ke_^mxs__
∧
∃c) : x∈α . ≡x . x = c} Df.
1 = α {(∃
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αihkj_^kl\hfh^gh-h^ghagZqgh]hhlghr_gbyι.
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b^ms_fmaZ αKe_^h\Zl_evgh^eyex[h]hdhg_qgh]hqbkeZdeZkkh\jZaebqguolbih\fu
fh`_fm\_ebqblv\k_bo^hlbiZdhlhjucfufh`_fgZa\ZlvgZbf_gvrbfh[sbffgh
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lZdbfkihkh[hfqlhj_amevlbjmxsb_deZkkug_[m^mlbf_lvh[sbowe_f_glh\AZl_ffu
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wlhlf_lh^ijbf_gblvg_evayIhwlhcijbqbg_fug_fh`_f^hdZaZlvqlh^he`gu[ulv
[_kdhg_qgu_deZkkuB[hij_^iheh`bfqlh[ueh[u\hh[s_lhevdhnbg^b\b^h\]^_n
–dhg_qghLh]^Z[ueh[u 2 n deZkkh\bg^b\b^h\ 2 2 deZkkh\deZkkh\bg^b\b^h\bl^LZ
dbfh[jZahfdZj^bgZevgh_qbkehqe_gh\dZ`^h]hlbiZ[ueh[udhg_qghbohlywlbqbk
eZij_\hkoh^beb[uex[h_aZ^Zggh_dhg_qgh_qbkehg_[ueh[ukihkh[Zkeh`blvbolZd
qlh[uihemqblv[_kdhg_qgh_qbkehKe_^h\Zl_evghgZfg_h[oh^bfZbih\k_c\b^bfh
klb lZd hgh b _klv ZdkbhfZ \ lhf kfuke_ qlhgbh^bg dhg_qguc deZkkbg^b\b^h\ g_
kh^_j`bl\k_bg^b\b^uh^gZdh_kebdlh-lhhl^Zklij_^ihql_gb_lhfmqlhh[s__qbkeh
bg^b\b^h\\mgb\_jkmf_jZ\ghkdZ`_flhih-\b^bfhfmg_lZijbhjgh]hkihkh[Z
hijh\_j]gmlv_]hfg_gb_
GZ hkgh\Zgbb ij_^eh`_ggh]h \ur_ kihkh[Z jZkkm`^_gby ykgh qlh ^hdljbgZ lbih\
ba[_]Z_l\k_oaZljm^g_gbchlghkbl_evghgZb[hevr_]hdZj^bgZevgh]hqbkeZGZb[hevr__
dZj^bgZevgh_qbkeh_klv\dZ`^hflbi_gh_]h\k_]^Zij_\hkoh^bldZj^bgZevgh_qbkeh
ke_^mxs_]hlbiZihkdhevdm_kebα –dZj^bgZevgh_qbkehh^gh]hlbiZlhdZj^bgZevgh_
qbkehke_^mxs_]hlbiZ_klv 2α dhlhjh_dZdihdZaZeDZglhj\k_]^Z[hevr_q_f αIh
kdhevdmg_kms_kl\m_lf_lh^Zkeh`_gbyjZaebqguolbih\fug_fh`_f]h\hjblvhµdZj
^bgZevghfqbke_\k_oh[t_dlh\dZdbo[ulhgb[uehlbih\¶bihwlhfmZ[khexlghgZb
[hevr_]hdZj^bgZevgh]hqbkeZg_kms_kl\m_l
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kms_kl\m_l deZkk 0 dZj^bgZevguo qbk_e Z bf_ggh deZkk dhg_qguo dZj^bgZevguo qb
k_eKe_^h\Zl_evgh 0bf__lf_klhdZddZj^bgZevgh_qbkehdeZkkZdeZkkh\deZkkh\bg
^b\b^h\H[jZamy\k_deZkkudhg_qguodZj^bgZevguoqbk_efugZoh^bfqlh 2ℵ bf__l
f_klhdZddZj^bgZevgh_qbkehdeZkkZdeZkkh\deZkkh\deZkkh\bg^b\b^h\blZdfufh
`_fijh^he`Zlvg_hij_^_ezggh^he]hFh`ghlZd`_^hdZaZlvkms_kl\h\Zgb_ n^eydZ
`^h]hdhg_qgh]hnghwlhlj_[m_ljZkkfhlj_gbyhj^bgZeh\
?keb\^h[Z\hddij_^iheh`_gbxqlhgbh^bgbadhg_qguodeZkkh\g_kh^_j`bl\k_o
bg^b\b^h\fuij_^iheZ]Z_ffmevlbiebdZlb\gmxZdkbhfml_Zdkbhfmqlh^eyaZ^Zg
gh]h fgh`_kl\Z \aZbfgh bkdexqZxsbo deZkkh\ gb h^bg ba dhlhjuo g_ y\ey_lky gme_
\uf _klv ih djZcg_c f_j_ h^bg deZkk \dexqZxsbc h^bg we_f_gl ba dZ`^h]h deZkkZ
wlh]hfgh`_kl\Zlhfufh`_f^hdZaZlvqlhkms_kl\m_ldeZkkkh^_j`Zsbc 0we_f_g
lh\ lZd qlh 0 [m^_l bf_lv f_klh dZd dZj^bgZevgh_ qbkeh bg^b\b^h\ Wlh g_kdhevdh
mf_gvrZ_llbi^hdhlhjh]hfu^he`gu^hclbqlh[u^hdZaZlvl_hj_fmhkms_kl\h\Zgbb
^eyex[h]haZ^Zggh]hdZj^bgZevgh]hqbkeZghg_^ZzlgZfdZdhc-eb[hl_hj_fuhkms_
kl\h\ZgbbdhlhjZyjZgvr_bebiha`_g_fh`_l[ulvihemq_gZbgZq_
Fgh]b_we_f_glZjgu_l_hj_fu\dexqZxsb_dZj^bgZevgu_qbkeZlj_[mxlfmevlbi
ebdZlb\gmx Zdkbhfm1 G_h[oh^bfh hlf_lblv qlh wlZ ZdkbhfZ wd\b\Ze_glgZ Zdkbhf_
P_jf_eh2bke_^h\Zl_evgh^hims_gbxqlhdZ`^ucdeZkkfh`_l[ulv\iheg_mihjy^h
n
0
Kj.: qZklv III fh_c klZlvb ‘On some Difficulties in the Theory of Transfinite Numbers and Order Types’,
Proc. London Math. Soc. Ser. II, Vol. IV, Part I.
2
H[Zdkbhf_P_jf_ehbh^hdZaZl_evkl\_lh]hqlhwlZZdkbhfZ\e_qzlfmevlbiebdZlb\gmxZdkbhfmkf
ij_^u^msmxkghkdmH[jZlguc\u\h^\u]ey^bllZdH[hagZqbfdZdProd‘kfmevlbiebdZlb\gucdeZkk
kjZkkfhljbf
1
q_g1Wlbwd\b\Ze_glgu_ij_^ihkuedbih-\b^bfhfm^hdZaZlvg_\hafh`ghg_kfhljygZ
lhqlhfmevlbiebdZlb\gZyZdkbhfZ\u]ey^bl^hklZlhqghijZ\^hih^h[ghc<hlkmlkl\bb
^hdZaZl_evkl\Z\b^bfhemqr_g_ijbgbfZlvfmevlbiebdZlb\gmxZdkbhfmdZd^hims_
gb_ghmklZgZ\eb\Zlv_zdZdmkeh\b_\dZ`^hfkemqZ_\dhlhjhfhgZbkihevam_lky
XHJ>BG:EVGU?QBKE:
Hj^bgZevgh_qbkeh_klvdeZkkhj^bgZevghkoh^guo\iheg_mihjy^hq_gguojy^h\l_
hlghr_gbch[jZamxsbolZdb_jy^uHj^bgZevgh_koh^kl\hbebih^h[b_hij_^_ey_lky
ke_^mxsbfh[jZahf
∧
∧
∪
∃S) . S∈1→1 . [D]‘S = C‘Q . P = SQ S }
Smor = P Q {(∃
Df.,
]^_µSmor¶_klvkhdjZs_gb_^eyµkoh^guhj^bgZevgh¶
DeZkkhlghr_gbcjy^Zdhlhju_fu[m^_fgZau\ZlvµSer¶hij_^_ey_lkylZd
∧
→
←
Ser = P {xPy . ⊃x, y . ∼ (x = y) : xPy . yPz . ⊃x, y, z . xPz : x∈ C‘P . ⊃x . P ‘x ∪ ι‘x ∪ P ‘x =
C‘P}Df.
L__kebqblZlvJdZdµij_^r_kl\m_l¶lhhlghr_gb_y\ey_lkyhlghr_gb_fjy^Z_keb
g_lgbh^gh]hwe_f_glZij_^r_kl\mxs_]hkZfhfmk_[_ij_^r_kl\_ggbdij_^r_
kl\_ggbdZ_klvij_^r_kl\_ggbd_kebo_klvdZdhc-lhqe_giheyhlghr_gbylhij_^
r_kl\_ggbdb o \f_kl_ k o \ kh\hdmighklb k _]h ij_^r_kl\_ggbdZfb h[jZamxl \kz ihe_
hlghr_gby
<iheg_mihjy^hq_ggu_hlghr_gbyjy^Zdhlhju_fu[m^_fgZau\ZlvΩhij_^_eyxl
kyke_^mxsbfh[jZahf
∧
∪
Ω = P {P∈ Ser : α ⊂ C‘P . ∃!α . ⊃α . ∃!(α – P ‘‘α)}
Df.;
l_ P ihjh`^Z_l \iheg_ mihjy^hq_ggu_ jy^u _keb J _klv hlghr_gb_ jy^Z b ex[hc
deZkk αkh^_j`Zsbcky\ihe_Jbg_y\eyxsbckygme_\ufbf__li_j\ucqe_gHlf_
lbfqlh P ‘‘αkmlvqe_gu\oh^ysb_ihke_g_dhlhjh]hqe_gZα).
?kebdZdNo‘Ph[hagZqblvhj^bgZevgh_qbkeh\iheg_mihjy^hq_ggh]hhlghr_gbyJZ
dZdNOdeZkkhj^bgZevguoqbk_elhfuihemqbf
∪
∧
∧
→
No = α P {P∈Ω . α = Smor ‘P} Df.,
NO = No‘‘Ω.
∧
bij_^iheh`bfqlh
∃x) . x∈β . D‘R = ι‘β . [D]‘R = ι‘x}
Z‘β = R {(∃
∧ ∧
Df.,
γ∈ Prod‘Z‘‘cl‘a . R = ξ x {(∃
∃S) . S∈γ . ξSx}.
Lh]^ZR –wlhkhhl\_lkl\b_P_jf_ehKe_^h\Zl_evgh_kebProd‘Z‘‘cl‘ag_y\ey_lkygme_\uflh^eyZ
kms_kl\m_lihdjZcg_cf_j_h^ghkhhl\_lkl\b_P_jf_eh
1
Kf.: Zermelo, ‘Beweis, dass jede Menge wohlgeordnet werden kann’. Math. Annalen, Vol. LIX, C.514-16.
Bahij_^_e_gbyNofuihemqZ_f
L: P∈ Ω . ⊃ . No‘P = Smor ‘P ,
L: ∼(P∈ Ω) . ⊃ . ∼E!No‘P.
→
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ih^mfZlvqlhhlghr_gb_ιihjh`^Z_ljy^hj^bgZevgh]hqbkeZωlbiZ
x, ι‘x, ι‘ι‘x, … ιn‘x, …,
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