Inconsistency of ℕ with the set union operation

Inconsistency of N with the set union operation Enrico P. G. Cadeddu ∗ 10 July 2025 Abstract A contradiction is obtained, considering the list of Nsub-chains, their inclusion relation and the set union operation. We discuss a possible simpler explanation and also we get a clear graphic-symbolic representation. Furthermore, inconsistency of Peano successor axiom is a consequence of rejecting infinity. Finally, in the conclusion section we get a proof about the inconsistency of infinity with a geometric description. Introduction The issue of infinity, in particular the actual infinity, leads us to write this article, as other previous ones [1] [2]. The purpose has always been to get a proof of inconsistency rather than hypothesize it in a new system, or arbitrarily deny infinity. We consider the axiom of infinity [4] [5] [8] , then the existence of N and Peano axioms [3]. Sets are considered with the usual graphical-symbolic notation {0,1,2, ...n}(see also [6] [7]). We start with the sets-list {x|x≤ y} ∀y∈Nwith all y taken together. Each set is shown to be finite, then using inclusion relation and union operation, we obtain a contradiction. We only consider actual infinity as the true infinity. Potential infinity should just be viewed as a growing finite. ∗Email Address [email protected] 1 1 Subsets, inclusion relation, set union operation and inconsistency of N We consider an infinite list of sets defined by: {x|x≤y} ∀y∈Nwith all y taken together (1) They are proper subsets of N. In agreement with the axiom of infinity and the axiom of separation, the set of all Natural numbers Nexists, together its subsets. We highlight the greatest number of each set, y, considering the list of these numbers (which are all numbers of N) together with (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 associate a different number to each set, for example as for Von Neumann number definition, but this is irrelevant about all considerations that follow; all numbers having to belong to the list in any case. Immediately we deduce that there isn’t a set equal to Nin the sets-list. In fact each y is a natural number (a finite number, as it is possible to demonstrate) that does have a successor for Peano axioms; so any set must not have all numbers, unlike N. So we have the 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, ....}. Or in a more compact form: I1⊂I2⊂I3⊂I4⊂I5⊂.... ⊂In⊂.... ⊂N(2) 2 Each set Iiis a proper subset of other following sets and obviously of N which is infinite, and each set Iihas one more number than the previous one. This chain is equivalent to this graphic-visual form of representation: Figure 1 It is very important to highlight: each set includes all preceding sets and has to contain some numbers not contained in any preceding set (at finite as well as infinity, this only depending on inclusion relation as in (2) and Figure 1). So, Nhas to contain at least some numbers not contained in any of its finite subsets Ii;Ncontains all numbers of all its subsets Iiand even more in agreement with the preceding proper subset symbol ”⊂” in (2). This aspect is completely solved in section 2 and specifically 2.1. Consequently, Iunion =Si∈NIi, the union of all subsets Ii, must not contain these numbers which are instead contained in N, and we have: Iunion ⊂N(3) But from (1), in which are contained all natural numbers in agreement with ∀y, also we have Si∈NIi=N, that is Iunion =Nand then: ¬(Iunion ⊂N)(4) So we have a contradiction, being simultaneously (3) and (4). 3 1.1 Another approach, a not-infinite Iunion Each Iiin (2) is a proper subset of Nas in (1), it is finite, each ”i” is finite. There aren’t infinite terms Ii. Then, on the left of ”⊂N” in (2) there would be a finite number of terms Ii, although indeterminate (for an y = n-1 the number of terms Iiis n). We further explain. The chain written in this manner: I1⊂I2⊂I3⊂ I4⊂...., containing all Natural numbers, cannot be considered consisting of infinite terms Ii, because all terms are finite, with a finite index ”i”, and then with a finite number of previous and subsequent terms; the number of terms between two terms is always finite. But if we write this: I1⊂I2⊂I3⊂ .... ⊂Iω(Iω=N), now an infinite number of terms can be exist, because the last term is infinite and then includes an infinite number of terms. We could think of an analogy with an open finite interval; for example the segment defined for [0,5) is not ”5” long but is less than ”5”. So, on the left of ”⊂N” in (2), there isn’t Iω, then the number of terms isn’t infinite (and each term is finite) and their union isn’t infinite. Even more clearly, if we had an infinite Iunion, then infinite quantity of numbers and an infinite chain, on the left of ”⊂N”, we would have infinite terms Iibut all finite, which only admit a finite chain (an Inis preceded by a finite number of Ii); there isn’t an infinite chain. This would be absurd. I1⊂I2⊂I3⊂... ⊂In⊂... /−−−∞Ii−−−−/It would be like locating a finite Inon infinity, that involves a finite chain, not infinite. In conclusion, Nis preceded, considering the ordering relation ”⊂”, by a finite set. A finite set is preceded by a finite number of sets, then a notinfinite chain. But this chain contains all Natural numbers, this is infinite and there is a contradiction. These considerations wouldn’t be valid if we thought to infinity as a finite chain increasing over time; in fact in this case the concept of infinity would be independent of the finiteness of every Ii. But time dependency cannot be taken into account to define actual infinity. We only have to consider time-independent statements. The axiom of infinity says us that all Iiexist simultaneously (there are all Ii) and we can’t add any more Ii. 4 1.2 A simplified and a first order approach A second-order logic has been used, because we have quantified on subsets. For a simplified approach and as an attempt at translation into the first order, we consider the chain: 0<1<2<3<4< .... < ω . Each Natural number n is finite and preceded by a finite number n of numbers. On the left of ”< ω” the chain 0<1<2<3<4< .... (that we call CN) isn’t infinite. In fact ωis greater than any Natural number and it is preceded, considering the ordering relation ”<”, by a Natural number (the chain only consists of Natural numbers), then a finite number. A finite number is preceded by a finite number of numbers, then the chain CNisn’t infinite. But it contains all Natural numbers and therefore it is the infinite set N, so we have a contradiction. Considering ωis necessary because Nis unlimited. ∀n∈CN(ω > n)−→?@x(x∈Ccard.∞)and ∀x(x∈Ccard.∞). Where Ccard.∞= a cardinal infinite chain. The difficult is to consider a generic finite number without a specific value n, to maintain Peano successor axiom. CNisn’t limited by a specific n, but it is limited by a finite number. These concepts seems to be outside the usual system of rules, but not wrong. ∀n∈ CN(ω > n)−→ @x(x≤n)(x∈Ccard.∞)is a solution? A rigid actual meaning (about time, considering actual infinity) implies CNis simply limited by a specific n (< ω) and all formal deductions, then contradictions, would fall within the normal rules, without the necessity of taking an unknown, undetermined number. Clearly invalidity of Peano successor axiom is an immediate consequence of the inferred finiteness of CN. ∀n∈CN(ω > n)−→ ∃M∀n(n≤M). M would be the finite number preceding ωby ”<”. This (contradictory) symbolic assertion (like the previous ones) has to be regarded as having a meta-theoretical meaning anyway, concerning the definition of the reference set (used for the theoretical symbolic calculation). Considering a successor of M, we would start a time-depending process, in contrast with the concept of ”actuality”. In practice Nshould be thought of as limited by a specific natural number, great enough to contain all necessary calculations. 5 2 Inconsistency in symbolic representation of N For ”actual” we mean that all numbers (and sub-sets ≡brackets) exist simultaneously (this is in (5), (6) and (7)). If we considered a bracket then another one and so on, without an end over time, we wouldn’t be in the context of actual infinity (or axiom of infinity). For the axiom of infinity, all numbers (and brackets) exist simultaneously and then nothing else can be added to these inside the set N. 2.1 We consider a set-representation of Nand its all proper subsets, all included between them, like in (1), (2) and Figure 1. Then: {{0},1},2},3},4},.... ? }, .... ?, .... }N(5) Blue brackets represent the sets of the list (1) (for simplicity we have only considered one blue left bracket). So, to describe the fact that all sets of the list are proper subsets of Nand simultaneously they observe relation (2), in the points ”}, ....?, ....}N” of (5) there are some numbers without blue brackets (if ”}N” was preceded by a blue bracket, there would be an improper subset); that is there are numbers of Nwhich don’t belong to any set of the list. This is a contradiction; any number have to belong to some set of the list. 2.2 Another approach is the following. We are directly referring to (1) and Si∈NIi=N. {{0},1},2},3},4},.... }}N(6) Nincludes all subsets (blue brackets) defined by list (1). These subsets contain each natural number y (as defined in (1)). In practice each number is accompanied by a blue bracket. Then we see a blue bracket including all natural numbers (and all subsets); the bracket ”}N” has to be preceded by a blue bracket, which includes all numbers preceding it. So, 6 there is a proper subset (a blue bracket) including all natural numbers (let’s keep in mind that there are all numbers). But no proper subset on the list (1) includes all numbers; each number having a successor (for Peano axioms) and then each subset having a ”successor-set”. So there is a contradiction; N⊂N. As already said (axiom of infinity) it is not possible to avoid a specific bracket ”}” immediately preceding ”}N”, adding another blue bracket before ”}N” and so on, over time. To complete, we see that, because (6) gives an absurd, the bracket ”}N” cannot be immediately preceded by a blue bracket (a proper subset including its numbers). So, to avoid this, Ncould be an empty set, but this is also absurd, a contradiction. 2.3 Finally, we have a direct connection with [2] considering a similar approach to (6), but without unnecessary N-brackets in this case. We take the union of all proper subsets in the list (1) again, continuing to specify them: {0}1}2}3}4}.... n}.... }(7) In (7) there are only brackets of proper subsets (blue brackets), there are all natural numbers and each number is accompanied by a blue bracket. So all natural numbers cannot be outside of a proper subset, implying N⊂N. This is a contradiction and an actual infinite list isn’t possible. In this section we have shown that sets graphic symbolic representation and evidences from (2) lead to contradictions. But, is sets symbolism so important? Anyway, this representation is in line with what’s said, above all in the first part of section 1. 3 Inconsistency of Peano successor axiom Supposing the set of Natural numbers is finite, there is a number, the greatest number, without a successor, in contrast with the Peano successor axiom (the successor could never be ”0”, for the other Peano axiom: zero is not the successor of any natural number). 7 So the set of Natural numbers isn’t finite, that is infinite. But this is in contrast to the inconsistency of infinity. Also an infinite time, infinite time intervals, couldn’t exist, and the process involving numbers that follow each other continuously would end. Then there is a contradiction. Conclusion Infinity (actual infinity) is a fundamental part of N, as set, that determines its existence. We have come, in the initial part of section 1 and in section 2, to affirm the inconsistency of actual infinity, which ultimately can be clarified as follows: Ndefines an infinite chain (also represented in a geometric place, e.g. a straight line) consisting of all and only finite numbers n (all natural numbers), each of which defining a finite chain (0,1,2, ....n) and then implying the presence of no infinite chain (in the same geometric place). In another way: the infinite chain, made up of all (and only) natural numbers, cannot coincide with any of its finite sub-chain (0,1,2, ....n), then it should contain some natural numbers mnot contained in any of those finite chains, but this isn’t possible by definition (any mdefining a finite chain) and there is a contradiction.Nchain: { 0,1,2,3,4, .... n, .... m, .... }N. There is no reference to the set union operation here. Furthermore it can be visualized by matching a chain to a segment defined from 0to x along the x-axis, and natural numbers being points of the segment at a finite distance ∆xfrom each other. The infinite segment (x=∞, but also a finite representation is possible with dx), being the longest, has to contain points (then natural numbers) not contained in any other finite segment. It is also observed that a natural number (then a finite number) at an infinite distance cannot define this distance itself. So, a geometric representation, specifically a ”number line” (which recalls the concept of chain), would seem necessary for the inconsistency of infinity. It should be noted that an ”external” consideration about chains (sets) is coherent. So the n+1 chain (0,1,2, ....n) doesn’t coincide with the n chain (0,1,2, ....n −1) because their external limits (n+1 and n) are different. At the same manner the ωchain, the chain of natural numbers, doesn’t coincide with any of its sub-chains because ωis different from any number n. However, 8 this consideration doesn’t eliminate the previous ones; the chain of all and only natural numbers has to be infinite (otherwise ωwould just be a fictitious symbol) and then contradictory. So, the chain of all natural numbers would be a finite chain; a chain with a finite number (but indeterminate) of terms. In section 2, repr. (5), (6) and (7), we have a clear description with set symbols about the inconsistency. Moreover, inconsistency of Peano successor axiom emerges as a consequence. All this leads us to ask some questions. What consequences might this inconsistency have on other theories that include N? Is it possible to speak about the existence of a local coherence, concerning time, with reference to Peano successor axiom? Is this inconsistency a demonstration of a concrete, finite physical reality? In our opinion the answer to this last question is affirmative. References [1] Enrico P G Cadeddu. Inconsistency of N and the question of infinity. OSF Preprints 10.31219/osf.io/2rs8u 2024 Jan. [2] Cadeddu Enrico P. G. Inconsistency of N from a not-finitist point of view. International Journal of Modern Research in Engineering and Technology (IJMRET), 8(10):15–16, 2023. [3] Giuseppe Peano. Arithmetices principia: Nova methodo exposita. Fratres Bocca, 1889. [4] Jerzy Pogonowski. “mathematics is the logic of the infinite”: Zermelo’s project of infinitary logic. Studies in Logic, Grammar and Rhetoric, 66(3):673–708, 2021. [5] Bertrand Russell. Introduction to mathematical philosophy. Taylor & Francis, 2022. [6] D Singh and JN Singh. von neumann universe: A perspective. International Journal of Contemporary Mathematical Sciences, 2:475–478, 2007. [7] John Von Neumann. Zur einfuhrung der transfiniten zahlen. Acta Litterarum ac Scientiarum Regiae Universitatis Hungaricae FranciscoJosephinae, sectio scientiarum mathematicarum, 1:199–208, 1923. 9