In e na ional Jou nal o Mode n Resea ch in Enginee ing and Technology (IJMRET)
www.ijm e .o g Volume 8 Issue 10 ǁ Decembe 2023.
w w w . i j m e . o g I S S N : 2 4 5 6 - 5 6 2 8
Page 15
Inconsis ency o ℕ om a no - ini is poin o iew
En ico Pie Gio gio Cadeddu
(Mas e o Science, O is ano, Sa dinia - I aly)
ABSTRACT: Conside ing he se o na u al numbe s
ℕ
, hen in he con ex o Peano axioms, s a ing om
inequali ies be ween ini e se s, we ind a undamen al con adic ion, abou he exis ence o
ℕ
, om a no - ini is
poin o iew.
KEYWORDS -Inconsis ency, Peano axioms, Na u al numbe s se , No - ini is
I.INTRODUCTION
A o mal sys em oge he an in e p e a ion,
cons i u ed o an alphabe , g amma , in e ence ules,
axioms, and a e e ence se , can p oduce o malized
p oposi ions and deduc ions ( heo ems) h ough wi h
a ini e numbe o s eps, ha is a ini is app oach [1,
2].
A sys em is consis en whe he a
p oposi ion and i s nega ion a e no deduced. Godel's
incomple eness heo ems [3], de eloped on he basis
o he sys em o P incipia Ma hema ica including he
axiom o in ini y, ep esen a o ess o logic and
consis ency agains inconsis ency. Bu a he same
ime hey ep esen a p elude o inconsis ency. They
gi e us necessa y condi ions o consis ency, no
su icien ones (undecidable p oposi ions and in e nal
no -demons able cohe ence a e hese necessa y
condi ions).
Conside ing he successo unc ion S(x) and
he exis ence o all na u al numbe s, in conco dance
wi h Peano axioms and he axiom o in ini y, we
show a con adic ion in
ℕ
, in a no - ini is way, ha
is hinking o ake all na u al numbe s
simul aneously.
II.NATURAL NUMBERS SET
The exis ence o
ℕ
is g an ed by he axiom
o in ini y [4, 5, 6]. This exis ence imply ha one o
each elemen o he se , also in an ac ual sense, so
aken all oge he . A ini e se wouldn' admi he
Peano axiom: ∀x(S(x)), wi h S(x) ∈
ℕ
, because he
g ea es numbe doesn' ha e a successo in o he
ini e se . All numbe s o
ℕ
a e de ined by Peano
axioms [7, 8, 9], oge he hei p op ie ies hanks o
he axiom o induc ion.
III.A FUNDAMENTAL CONTRADICTION
The wo se s: {0, {S(x) | x ∈
ℕ
}} (wi h S(x)
∈
ℕ
) and
ℕ
, a e he same se , ha is:
{0, {S(x) | x ∈
ℕ
}} = {x | x ∈
ℕ
} =
ℕ
(1)
We know, as i is demons able, ha : (x ∈
ℕ
)(∀x(x < S(x)). Tha is 0 < 1, 1< 2, …, n < n+1.
A he same ime we ha e:
{x | x ≤ y} ≠ {x | x ≤ y + 1} ∀y
(2)
wi h y+1 = S(y) ∈
ℕ
.
Tha is {0, 1, 2, 3} ≠ {0, 1, 2, 3, 4} and so
on, o all y.
Bu necessa y condi ion o ha e all y ( ha is
∀y) is ha a leas one o all hese se s in (2) exis s
equal o
ℕ
, o he wise all y a e no aken; he absence
o
ℕ
(all numbe s) in (2) would imply ha we could
add numbe s no p esen in each se in (2) (so, many
numbe s would be absen in each se ). Then,
conside ing all y, hen all x, and equa ion (1), we a e
conside ing in (2) a se equal o
ℕ
. So we ha e
ℕ
≠
ℕ
,
a con adic ion.
I is o no ice a ques ion: is i necessa y o
pass h ough a necessa y condi ion o , di ec ly, do all
y imply a se equal o
ℕ
? A i s sigh he answe
seems no and yes espec i ely.
CONCLUSION
This p oo o inconsis ency is no - ini is
because i in ol es in ini e o ali ies. Bu his is
na u al conside ing he se heo y wi h he axiom o
in ini y (all elemen s o
ℕ
). On he o he hand a
ini is p oo would imply he end o ma hema ics as
we know i .
Anyway, e using a p ecise de ini ion o
ℕ
,
hen e using he axiom o in ini y (and Peano
In e na ional Jou nal o Mode n Resea ch in Enginee ing and Technology (IJMRET)
www.ijm e .o g Volume 8 Issue 10 ǁ Decembe 2023.
w w w . i j m e . o g I S S N : 2 4 5 6 - 5 6 2 8
Page 16
axioms?), could be a iew o a oid his
inconsis ency. So he axiom o in ini y would seem
o ha e a simila ole o cohe ence. I is no
demons able, bu also i canno be aken as an axiom
i one doesn' wan a sys em o be inconsis en . This
p oo suppo s ini is app oach in a no a bi a y
manne and all heo ies implying
ℕ
wi h he axiom o
in ini y could be e isi ed (including Godel’s
heo ems).
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