Inconsis ency o N
wi h he se union ope a ion
En ico P. G. Cadeddu ∗
10 July 2025
Abs ac
A con adic ion is ob ained, conside ing he lis o Nsub-chains,
hei inclusion ela ion and he se union ope a ion. We discuss a
possible simple explana ion and also we ge a clea g aphic-symbolic
ep esen a ion. Fu he mo e, inconsis ency o Peano successo axiom
is a consequence o ejec ing in ini y. Finally, in he conclusion sec ion
we ge a p oo abou he inconsis ency o in ini y wi h a geome ic
desc ip ion.
In oduc ion
The issue o in ini y, in pa icula he ac ual in ini y, leads us o w i e his
a icle, as o he p e ious ones [1] [2]. The pu pose has always been o ge a
p oo o inconsis ency a he han hypo hesize i in a new sys em, o a bi-
a ily deny in ini y.
We conside he axiom o in ini y [4] [5] [8] , hen he exis ence o N
and Peano axioms [3]. Se s a e conside ed wi h he usual g aphical-symbolic
no a ion {0,1,2, ...n}(see also [6] [7]). We s a wi h he se s-lis {x|x≤
y} ∀y∈Nwi h all y aken oge he . Each se is shown o be ini e,
hen using inclusion ela ion and union ope a ion, we ob ain a con adic ion.
We only conside ac ual in ini y as he ue in ini y. Po en ial in ini y
should jus be iewed as a g owing ini e.
∗Email Add ess [email p o ec ed]
1
1 Subse s, inclusion ela ion, se union ope -
a ion and inconsis ency o N
We conside an in ini e lis o se s de ined by:
{x|x≤y} ∀y∈Nwi h all y aken oge he (1)
They a e p ope subse s o N. In ag eemen wi h he axiom o in ini y
and he axiom o sepa a ion, he se o all Na u al numbe s Nexis s, oge he
i s subse s.
We highligh he g ea es numbe o each se , y, conside ing he lis o
hese numbe s (which a e all numbe s o N) oge he wi h (1):
{0} 0
{0,1} 1
{0,1,2} 2
{0,1,2,3} 3
{0,1,2,3,4} 4
..
..
..
{ 0, 1, 2, 3, 4, ... ? } ?
Also we can associa e a di e en numbe o each se , o example as o
Von Neumann numbe de ini ion, bu his is i ele an abou all conside a-
ions ha ollow; all numbe s ha ing o belong o he lis in any case. Imme-
dia ely we deduce ha he e isn’ a se equal o Nin he se s-lis . In ac each
y is a na u al numbe (a ini e numbe , as i is possible o demons a e) ha
does ha e a successo o Peano axioms; so any se mus no ha e all num-
be s, unlike N. So we ha e he chain: {0} ⊂ {0,1} ⊂ {0,1,2} ⊂ {0,1,2,3} ⊂
{0,1,2,3,4} ⊂ .... ⊂ {0,1,2,3,4, ....n −1} ⊂ .... ⊂ {0,1,2,3,4, ....}. O in a
mo e compac o m:
I1⊂I2⊂I3⊂I4⊂I5⊂.... ⊂In⊂.... ⊂N(2)
2
Each se Iiis a p ope subse o o he ollowing se s and ob iously o N
which is in ini e, and each se Iihas one mo e numbe han he p e ious one.
This chain is equi alen o his g aphic- isual o m o ep esen a ion:
Figu e 1
I is e y impo an o highligh : each se includes all p eceding se s
and has o con ain some numbe s no con ained in any p eceding
se (a ini e as well as in ini y, his only depending on inclusion
ela ion as in (2) and Figu e 1).
So, Nhas o con ain a leas some numbe s no con ained in any o i s
ini e subse s Ii;Ncon ains all numbe s o all i s subse s Iiand e en mo e in
ag eemen wi h he p eceding p ope subse symbol ”⊂” in (2). This aspec
is comple ely sol ed in sec ion 2 and speci ically 2.1.
Consequen ly, Iunion =Si∈NIi, he union o all subse s Ii, mus no con ain
hese numbe s which a e ins ead con ained in N, and we ha e:
Iunion ⊂N(3)
Bu om (1), in which a e con ained all na u al numbe s in ag eemen
wi h ∀y, also we ha e Si∈NIi=N, ha is Iunion =Nand hen:
¬(Iunion ⊂N)(4)
So we ha e a con adic ion, being simul aneously (3) and (4).
3
1.1 Ano he app oach, a no -in ini e Iunion
Each Iiin (2) is a p ope subse o Nas in (1), i is ini e, each ”i” is ini e.
The e a en’ in ini e e ms Ii. Then, on he le o ”⊂N” in (2) he e would
be a ini e numbe o e ms Ii, al hough inde e mina e ( o an y = n-1 he
numbe o e ms Iiis n).
We u he explain. The chain w i en in his manne : I1⊂I2⊂I3⊂
I4⊂...., con aining all Na u al numbe s, canno be conside ed consis ing o
in ini e e ms Ii, because all e ms a e ini e, wi h a ini e index ”i”, and hen
wi h a ini e numbe o p e ious and subsequen e ms; he numbe o e ms
be ween wo e ms is always ini e. Bu i we w i e his: I1⊂I2⊂I3⊂
.... ⊂Iω(Iω=N), now an in ini e numbe o e ms can be exis , because he
las e m is in ini e and hen includes an in ini e numbe o e ms. We could
hink o an analogy wi h an open ini e in e al; o example he segmen
de ined o [0,5) is no ”5” long bu is less han ”5”. So, on he le o ”⊂N”
in (2), he e isn’ Iω, hen he numbe o e ms isn’ in ini e (and each e m
is ini e) and hei union isn’ in ini e.
E en mo e clea ly, i we had an in ini e Iunion, hen in ini e quan i y o
numbe s and an in ini e chain, on he le o ”⊂N”, we would ha e in ini e
e ms Iibu all ini e, which only admi a ini e chain (an Inis p eceded by
a ini e numbe o Ii); he e isn’ an in ini e chain. This would be absu d.
I1⊂I2⊂I3⊂... ⊂In⊂...
/−−−∞Ii−−−−/I would be like loca ing a ini e Inon in ini y,
ha in ol es a ini e chain, no in ini e.
In conclusion, Nis p eceded, conside ing he o de ing ela ion ”⊂”, by
a ini e se . A ini e se is p eceded by a ini e numbe o se s, hen a no -
in ini e chain. Bu his chain con ains all Na u al numbe s, his is in ini e
and he e is a con adic ion.
These conside a ions wouldn’ be alid i we hough o in ini y
as a ini e chain inc easing o e ime; in ac in his case he concep
o in ini y would be independen o he ini eness o e e y Ii. Bu
ime dependency canno be aken in o accoun o de ine ac ual
in ini y. We only ha e o conside ime-independen s a emen s.
The axiom o in ini y says us ha all Iiexis simul aneously ( he e
a e all Ii) and we can’ add any mo e Ii.
4
1.2 A simpli ied and a i s o de app oach
A second-o de logic has been used, because we ha e quan i ied on subse s.
Fo a simpli ied app oach and as an a emp a ansla ion in o he i s
o de , we conside he chain:
0<1<2<3<4< .... < ω .
Each Na u al numbe n is ini e and p eceded by a ini e numbe n o
numbe s. On he le o ”< ω” he chain 0<1<2<3<4< .... ( ha
we call CN) isn’ in ini e. In ac ωis g ea e han any Na u al numbe and
i is p eceded, conside ing he o de ing ela ion ”<”, by a Na u al numbe
( he chain only consis s o Na u al numbe s), hen a ini e numbe . A ini e
numbe is p eceded by a ini e numbe o numbe s, hen he chain CNisn’
in ini e. Bu i con ains all Na u al numbe s and he e o e i is he in ini e
se N, so we ha e a con adic ion. Conside ing ωis necessa y because Nis
unlimi ed.
∀n∈CN(ω > n)−→?@x(x∈Cca d.∞)and ∀x(x∈Cca d.∞). Whe e
Cca d.∞= a ca dinal in ini e chain. The di icul is o conside a gene ic
ini e numbe wi hou a speci ic alue n, o main ain Peano successo axiom.
CNisn’ limi ed by a speci ic n, bu i is limi ed by a ini e numbe . These
concep s seems o be ou side he usual sys em o ules, bu no w ong. ∀n∈
CN(ω > n)−→ @x(x≤n)(x∈Cca d.∞)is a solu ion?
A igid ac ual meaning (abou ime, conside ing ac ual in in-
i y) implies CNis simply limi ed by a speci ic n (< ω) and all o -
mal deduc ions, hen con adic ions, would all wi hin he no mal
ules, wi hou he necessi y o aking an unknown, unde e mined
numbe . Clea ly in alidi y o Peano successo axiom is an immedia e con-
sequence o he in e ed ini eness o CN.
∀n∈CN(ω > n)−→ ∃M∀n(n≤M). M would be he ini e numbe p e-
ceding ωby ”<”. This (con adic o y) symbolic asse ion (like he p e ious
ones) has o be ega ded as ha ing a me a- heo e ical meaning anyway, con-
ce ning he de ini ion o he e e ence se (used o he heo e ical symbolic
calcula ion). Conside ing a successo o M, we would s a a ime-depending
p ocess, in con as wi h he concep o ”ac uali y”. In p ac ice Nshould be
hough o as limi ed by a speci ic na u al numbe , g ea enough o con ain
all necessa y calcula ions.
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2 Inconsis ency in symbolic
ep esen a ion o N
Fo ”ac ual” we mean ha all numbe s (and sub-se s ≡b acke s) exis si-
mul aneously ( his is in (5), (6) and (7)). I we conside ed a b acke hen
ano he one and so on, wi hou an end o e ime, we wouldn’ be in he
con ex o ac ual in ini y (o axiom o in ini y).
Fo he axiom o in ini y, all numbe s (and b acke s) exis simul aneously
and hen no hing else can be added o hese inside he se N.
2.1
We conside a se - ep esen a ion o Nand i s all p ope subse s, all included
be ween hem, like in (1), (2) and Figu e 1. Then:
{{0},1},2},3},4},.... ? }, .... ?, .... }N(5)
Blue b acke s ep esen he se s o he lis (1) ( o simplici y we ha e only
conside ed one blue le b acke ).
So, o desc ibe he ac ha all se s o he lis a e p ope sub-
se s o Nand simul aneously hey obse e ela ion (2), in he poin s
”}, ....?, ....}N” o (5) he e a e some numbe s wi hou blue b acke s (i ”}N”
was p eceded by a blue b acke , he e would be an imp ope subse ); ha is
he e a e numbe s o Nwhich don’ belong o any se o he lis . This is a
con adic ion; any numbe ha e o belong o some se o he lis .
2.2
Ano he app oach is he ollowing.
We a e di ec ly e e ing o (1) and Si∈NIi=N.
{{0},1},2},3},4},.... }}N(6)
Nincludes all subse s (blue b acke s) de ined by lis (1). These subse s
con ain each na u al numbe y (as de ined in (1)). In p ac ice each numbe
is accompanied by a blue b acke . Then we see a blue b acke including
all na u al numbe s (and all subse s); he b acke ”}N” has o be
p eceded by a blue b acke , which includes all numbe s p eceding i . So,
6
he e is a p ope subse (a blue b acke ) including all na u al numbe s (le ’s
keep in mind ha he e a e all numbe s).
Bu no p ope subse on he lis (1) includes all numbe s; each num-
be ha ing a successo ( o Peano axioms) and hen each subse ha ing a
”successo -se ”. So he e is a con adic ion; N⊂N.
As al eady said (axiom o in ini y) i is no possible o a oid a speci ic b acke
”}” immedia ely p eceding ”}N”, adding ano he blue b acke be o e ”}N” and
so on, o e ime.
To comple e, we see ha , because (6) gi es an absu d, he b acke ”}N” can-
no be immedia ely p eceded by a blue b acke (a p ope subse including
i s numbe s). So, o a oid his, Ncould be an emp y se , bu his is also
absu d, a con adic ion.
2.3
Finally, we ha e a di ec connec ion wi h [2] conside ing a simila app oach
o (6), bu wi hou unnecessa y N-b acke s in his case.
We ake he union o all p ope subse s in he lis (1) again, con inuing
o speci y hem:
{0}1}2}3}4}.... n}.... }(7)
In (7) he e a e only b acke s o p ope subse s (blue b acke s), he e a e all
na u al numbe s and each numbe is accompanied by a blue b acke . So all
na u al numbe s canno be ou side o a p ope subse , implying
N⊂N. This is a con adic ion and an ac ual in ini e lis isn’ possible.
In his sec ion we ha e shown ha se s g aphic symbolic ep esen a ion and
e idences om (2) lead o con adic ions. Bu , is se s symbolism so impo -
an ? Anyway, his ep esen a ion is in line wi h wha ’s said, abo e all in
he i s pa o sec ion 1.
3 Inconsis ency o Peano successo axiom
Supposing he se o Na u al numbe s is ini e, he e is a numbe , he g ea es
numbe , wi hou a successo , in con as wi h he Peano successo axiom
( he successo could ne e be ”0”, o he o he Peano axiom: ze o is no he
successo o any na u al numbe ).
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So he se o Na u al numbe s isn’ ini e, ha is in ini e.
Bu his is in con as o he inconsis ency o in ini y.
Also an in ini e ime, in ini e ime in e als, couldn’ exis , and he p ocess
in ol ing numbe s ha ollow each o he con inuously would end. Then he e
is a con adic ion.
Conclusion
In ini y (ac ual in ini y) is a undamen al pa o N, as se , ha de e mines i s
exis ence. We ha e come, in he ini ial pa o sec ion 1 and in sec ion 2, o
a i m he inconsis ency o ac ual in ini y, which ul ima ely can be cla i ied as
ollows: Nde ines an in ini e chain (also ep esen ed in a geome ic
place, e.g. a s aigh line) consis ing o all and only ini e num-
be s n (all na u al numbe s), each o which de ining a ini e chain
(0,1,2, ....n) and hen implying he p esence o no in ini e chain (in
he same geome ic place).
In ano he way: he in ini e chain, made up o all (and only)
na u al numbe s, canno coincide wi h any o i s ini e sub-chain
(0,1,2, ....n), hen i should con ain some na u al numbe s mno
con ained in any o hose ini e chains, bu his isn’ possible by
de ini ion (any mde ining a ini e chain) and he e is a con adic-
ion.Nchain: { 0,1,2,3,4, .... n, .... m, .... }N.
The e is no e e ence o he se union ope a ion he e. Fu he mo e i can
be isualized by ma ching a chain o a segmen de ined om 0 o x along
he x-axis, and na u al numbe s being poin s o he segmen a a ini e dis-
ance ∆x om each o he . The in ini e segmen (x=∞, bu also a ini e
ep esen a ion is possible wi h dx), being he longes , has o con ain poin s
( hen na u al numbe s) no con ained in any o he ini e segmen . I is also
obse ed ha a na u al numbe ( hen a ini e numbe ) a an in ini e dis ance
canno de ine his dis ance i sel . So, a geome ic ep esen a ion, speci ically
a ”numbe line” (which ecalls he concep o chain), would seem necessa y
o he inconsis ency o in ini y.
I should be no ed ha an ”ex e nal” conside a ion abou chains (se s)
is cohe en . So he n+1 chain (0,1,2, ....n) doesn’ coincide wi h he n chain
(0,1,2, ....n −1) because hei ex e nal limi s (n+1 and n) a e di e en . A
he same manne he ωchain, he chain o na u al numbe s, doesn’ coincide
wi h any o i s sub-chains because ωis di e en om any numbe n. Howe e ,
8
his conside a ion doesn’ elimina e he p e ious ones; he chain o all and
only na u al numbe s has o be in ini e (o he wise ωwould jus be a ic i ious
symbol) and hen con adic o y.
So, he chain o all na u al numbe s would be a ini e chain; a chain wi h a
ini e numbe (bu inde e mina e) o e ms. In sec ion 2, ep . (5), (6) and
(7), we ha e a clea desc ip ion wi h se symbols abou he inconsis ency.
Mo eo e , inconsis ency o Peano successo axiom eme ges as a consequence.
All his leads us o ask some ques ions.
Wha consequences migh his inconsis ency ha e on o he heo ies ha
include N?
Is i possible o speak abou he exis ence o a local cohe ence, conce ning
ime, wi h e e ence o Peano successo axiom?
Is his inconsis ency a demons a ion o a conc e e, ini e physical eali y?
In ou opinion he answe o his las ques ion is a i ma i e.
Re e ences
[1] En ico P G Cadeddu. Inconsis ency o N and he ques ion o in ini y.
OSF P ep in s 10.31219/os .io/2 s8u 2024 Jan.
[2] Cadeddu En ico P. G. Inconsis ency o N om a no - ini is poin o iew.
In e na ional Jou nal o Mode n Resea ch in Enginee ing and Technology
(IJMRET), 8(10):15–16, 2023.
[3] Giuseppe Peano. A i hme ices p incipia: No a me hodo exposi a. F a es
Bocca, 1889.
[4] Je zy Pogonowski. “ma hema ics is he logic o he in ini e”: Ze melo’s
p ojec o in ini a y logic. S udies in Logic, G amma and Rhe o ic,
66(3):673–708, 2021.
[5] Be and Russell. In oduc ion o ma hema ical philosophy. Taylo &
F ancis, 2022.
[6] D Singh and JN Singh. on neumann uni e se: A pe spec i e. In e na-
ional Jou nal o Con empo a y Ma hema ical Sciences, 2:475–478, 2007.
[7] John Von Neumann. Zu ein uh ung de ans ini en zahlen. Ac a
Li e a um ac Scien ia um Regiae Uni e si a is Hunga icae F ancisco-
Josephinae, sec io scien ia um ma hema ica um, 1:199–208, 1923.
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