Inconsis ency o In ini y in a geome ic con ex
En ico P. G. Cadeddu
*
28 Sep embe 2025
Abs ac
Rep esen a ion o Nand hen i s ini e sub-chains along a line-segmen (o
a line) leads o a con adic ion conce ning ac ual in ini y; he longes line-
segmen , co esponding o N, con ains some na u al numbe s no con ained in
any sho e line-segmen s co esponding o all sub-chains.
Inconsis ency p oo
This p oo o inconsis ency, ega ding he ac ual in ini y hen he se o all na u al
numbe s, has al eady been ea ed [1], e en i in a less de ailed way on he geome ic
aspec .
I1⊂I2⊂I3⊂I4⊂I5⊂.... ⊂In⊂.... ⊂N(1)
Each ini e se (o chain) Ii, wi h he o m {0,1,2,3,4, ....i −1}, has one mo e
numbe han he p e ious one. I is a p ope subse o o he ollowing se s and
ob iously o Nwhich is in ini e. I mus be: Si∈NIi=N.
We ake a ini e line-segmen in which he chain N≡ {0,1,2,3,4, ....}o all na u al
numbe s is ep esen ed; a numbe is iden i ied wi h a poin and he dis ance be ween
wo poin s is dx, an in ini esimal dis ance (Figu e 1).
Figu e 1
In his line-segmen we ha e he in ini e chain o na u al numbe s and hen all
i s ini e p ope sub-chains Ii≡ {0,1,2,3,4, ...i −1}.ω /∈Nand delimi s he in ini e
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chain N.
E e y ini e Iicon ains less numbe s han N, which con ains all na u al numbe s,
see ela ion (1). Then in ini e chain N, co esponding o he en i e line-segmen ,
co esponds o he longes segmen and ini e sub-chains Iico espond o sho e
segmen s. Clea ly Ncanno coincide wi h any p ope sub-chain Ii.
So his en i e segmen , co esponding o N, con ains some poin s, hen
some na u al numbe s, no con ained in any o he sho e segmen co e-
sponding o a ini e sub-chain Ii. Bu his isn’ possible by de ini ion, because
any na u al numbe (o N) has o belong o a ini e sub-chain Iiand hen con ained
in a co esponding segmen sho e han he en i e segmen . The e is a con adic ion.
Taking a ini e line-segmen isn’ a necessa y condi ion o he p oo . We can ake a
line (an in ini e segmen ) wi h a ini e dis ance ∆xbe ween wo poin s co esponding
o wo na u al numbe s. The esul is he same: he line con ains some poin s, hen
some na u al numbe s, no con ained in any o he segmen co esponding o a ini e
sub-chain, which gi es a con adic ion. Bu in his case we also see ha he line,
co esponding o he in ini e chain N, has some poin s, hen some na u al numbe s,
a in ini y. This implies a na u al numbe (which is ini e in any case) de ining an
in ini e dis ance, an impossible s a emen , n·∆xbeing a ini e line-segmen ∀n, hen
a con adic ion.
This p oo was achie ed because all segmen s lie along he same segmen (o line) and
a compa ison can be made, in pa icula we can see ha he longes segmen has o
con ain some poin s, hen some numbe s, no con ained in any o he segmen . F om
his we see he impo ance o a nume ical-geome ic app oach. A pu ely nume ical
app oach o a pu ely geome ic one appea s no su icien o ob ain an inconsis ency
p oo .
I should also be emphasized ha inconsis ency o ac ual in ini y implies inconsis ency
o in ini esimals (a line-segmen would con ain an in ini e amoun o in ini esimals,
hen a con adic ion).
Geome y is di ec ly connec ed o space, which in u n is connec ed o physical eali y,
hen his p oo seems o a i m he ini eness o physical eali y.
Re e ences
[1] En ico P G Cadeddu. Inconsis ency o N wi h he se union ope a ion. Zenodo
h ps://doi.o g/10.5281/zenodo.10530599 2024-2025.
2
