# Are there theories of arithmetic that are inconsistent with the natural numbers?

The programme of ultrfinitism dispenses with the notion of very large finite numbers simply becaause they argue that such large finite numbers have no way of being conceptualised in our universe in a constructive manner.

As far as I can gather, the programme hasn't developed a sufficiently precise theory.

This obviously is inconsistent with the existence of the natural numbers as we usually know them.

Are there other theories of arithmetic that are inconsistent with the arithemetic as encodedin the Peano Axioms?

• I'm not sure that you can find "really interesting" results about this topic. The issue with set theory is different; also Intuitionism, the most developed "alternative" approach to classical math accept the infinity of natural numbers, at least in term of unlimited iteration of the process of "+1" (i.e.aristotelian potential infinite versus actual infinite). Commented Feb 6, 2014 at 10:25
• Point taken. If one considers the heirarchy of large cardinals, one could as well say that one has simply appended the weakest infinite axiom. ultrafinitism is an 'extreme' view; still there is a paraconsistent theory PA# of PA which has a finite model and that can be represented in the theory itself. And hence has a finatary consistency proof formalizable in PA#. Thats a reasonably interesting result. See Meyer. Commented Feb 6, 2014 at 10:37
• As said in other comment, there are a lot of research on "alternative" (logic,set theory,etc.) but I personally think that, apart from the interest in "intelelctual speculation", there is no value in "throwing away" natural numbers. 2,5 millenia of math has found no contradiction in them (compare set theory: in a couple of decades, math community was able to find the flaw of Cantor's and Frege's systems ...) and the (potential) infinity of the succession of natural numbers is so "deeply" radicated in our thought and language, that I think it is the clue of something really there. Commented Feb 6, 2014 at 10:50
• @allegranza: Sure, I don't disagree there; but that shouldn't stop one from speculating that there maybe systems of arithmetic where very large numbers are essentially the same. Though quite how, formally, speaking, I don't know. This speculation is hardly going to derail the mainstream tradition, because, as you say, its deeply embedded in our psyche. Commented Mar 19, 2014 at 6:24
• – Drux
Commented Mar 19, 2014 at 10:45

I suspect that you will not find an interesting mathematical theory about finite arithmetic.

Ultrafinitism is a consistent and interesting philosophical theory (I mean : philosophy of mathematics), but what kind of arithmetical "interesting" facts do you expect to find in such a theory ?

If we simply delete the Peano axiom which states that for every number there is a successor, we have the "finitist" Peano arithmetic.

But we still want all other properties of numbers ? If we work with "little" numbers, their sum and product will be still "little"; so, no problem.

But what will happen if we add two "big" numbers, i.e.two finite numbers that are quite at the "border" of the "biggest" thinkable or computable number ? We will step outside the "domain" ? or we have to enlarge it ?

We have no support for the infinitude of our physical world; thus, the infinity of the natural numbers may be only a fiction. But it is still a very very useful and interesting fiction...

• What do you mean by "interesting"? Commented Mar 22, 2014 at 19:26
• @SniperClown - If we put an "upper limit" to the "thinkable" or "practicaly usable" numbers rejecting the relevant Peano axiom, we will have the domain of the "finite" numbers. What properties they will have, in addition to "being little" ? Commented Mar 22, 2014 at 19:58
• @allegranza: I think its an interesting idea, I wouldn't call it a philosophy or a theory yet, if in fact it ever does attain that status. Commented Mar 23, 2014 at 10:24
• @allegranza: see my, or rather Hamkins answer below; an interesting twist on ultrafinitism is that he considers lack of closure under exponentiation. Commented Jun 8, 2014 at 4:20
• @MauroALLEGRANZA I dabbled briefly with a formalization of finite arithmetic, introducing a maximum number that has no successor. I got so far as to develop a partial function for addition, but things quickly bogged down after that. With all the special cases (the overflow conditions to use the CS jargon), it is probably not possible to go much beyond that, even to proving associativity. (See the entry "Arithmetic on Finite Sets" dated April 23, 2014 at my blog at dcproof.wordpress.com for a construction of + relation) Commented Jun 9, 2014 at 16:44

The thing is that in First Order Logic, you can have consistent theories that state "The size of the universe is cardinality N" as an axiom.

Trouble is, for those axioms, it is also a consistent theory to replace "N" with "N+1".

Also, the moment you do away with the axiom, and the universe is infinite in size, you cannot control how infinite it is.

There are some models of PA which are much larger than the real number-line as classically constructed; there are models of ZFC which are countable.

Ultrafinitism is a very strange thing indeed. It is kind of like Hipsters. They disregard something that obviously works, in favour of a "novel" idea.

• Well, I'm not an ultra-finitist. But what they are questioning is intriguing - that the universe is not large enough that we can write down every finite number. Commented Mar 23, 2014 at 10:22
• @MoziburUllah To which every editor of the Gogology Wiki will smack their forehead and say "Duh!" It also disregards that such large numbers are useful: Grahams Ordinal, for instance, has been used in a serious proof. It is so large that the remaining 348 characters fail me. Commented Mar 23, 2014 at 10:31
• Well perhaps some ultra-finitist extremists think that, but I don't see why this means that they are mutually exclusive. Or are you saying exactly that? One can be quite well aware that large ordinals are used to describe proof-theoretic strengths without thinking that this means that other options are thus invalidated. Finitism has yielded results elsewhere, QM in its standard formulation uses infinite-dimensional Fock Space, but there is finite QM. Commented Mar 23, 2014 at 11:09
• If we add to Peano axioms (without the axiom which states that for every number there is a successor) a f-o formula saying that there are at most N "objects" (i.e. N numbers) we may have the following "equation" : (N-1) +2 = (N-1) + S(1) = S(N-1) + 1 = N + S(0) = S(N) + 0 = undefined (like division by 0) because S(N) does not exists. Apart this "obviously true" fact in a universe with at most N objects, if we add n,m < N/2, their sum will be still n+m < N, and all works as usual... This is way I say that it is not interesting. Commented Mar 24, 2014 at 11:01
• @MoziburUllah Well, if one identifies with the Ultrafinitism-crowd but at the same time recognizes that unphysically large numbers are useful; what exactly separates an ultrafinitist from a regular, infinite-accepting mathematician? Suddenly it seems more like a special-snowflake mathematician in-crowd who think large numbers are "bad," for some value of bad-ness. Commented Mar 24, 2014 at 11:23

a set theorist, in his paper On Ontology & Realism in Mathematics writes:

To be more specific, the ultrafinitist basic position is that the natural numbers are closed under addition and multiplication, but are not closed under exponentiation...The choice of exponentiation, rather than some other fast growing function, seems right. It marks the first crucial big jump: in computer science ─ from polynomial time to exponential time, and in set theory ─ from a set to its power set (which is even more striking for infinite sets than for finite ones)

The position originated in the works of Esenin-Volpin [EV]... which are often obscure but contain some striking suggestive ideas. A major step was achieved by Parikh [P]. Dummett [D] gave an intuitive semi-formal argument purporting to show that ultrafinitism is incoherent; but the argument fails in an interesting way. Nelson’s book [N], in spite of its faults, is impressive in its systematic rigorous working out of a formal deductive system.

Although ultrafinitism is a topic of lively discussion on the internet, Nelson’s book is, as far as I know, the only worked out attempt at a full fledged formal system.

References:

[EV] : Esenin-Volpin, A. (1961) A “Le programme ultra-intuitionniste des fondements des mathématiques” In Infinitistic Methods, Proceedings of the Symposium on the Foundations of Mathematics, pages 201-223. Warsaw, 1961.

[P] :Parikh, R. (1971) “Existence and Feasibility in Arithmetic”, The Journal of Symbolic Logic 36, pp. 494–508.

[D] :Dummett, M. (1975) “Wang’s Paradox” Synthese 30, pp. 301-324.

[N] : Nelson, (1986) Predicative Arithmetic, Mathematical Notes 12, Princeton University Press. No. 3 pp. 329-353

• Sounds like quite a hodgepodge with closure under addition and multiplication but not under exponentiation. Haven't read the book (not likely to), but did the author manage to even develop the laws of exponents? Commented Jun 9, 2014 at 16:59
• Intuitionism was a hodge-pode when Brouwer invented it in opposition to Hilberts formalism and Cantorian Set Theory; its taken a century for it to attain mains-stream intellectual respectability; The calculus was a Hodge-podge when it was invented by Newton & Leibniz as remarked by Bishop Berkeley, which spurred mathematicians like Cauchy, Weierstrass, Robinson & Lawvere to address those critiques; beginnings are always something of a hodge-podge, its only the backward historical look that constructs a narrative that gives it due clarity; Commented Jun 9, 2014 at 23:57
• Ultra-finitism may or may not work out, but I have enough respect for Hamkins opinion to give it due regard, plus the fact it is in itself an interesting question; I'm not intending to read the book either - but I don't expect that he would develop the law of exponents if he isn't going to show closure; the real question is what is the right formal context for this, and is there one; Nelson has made a beginning, perhaps others will follow. Commented Jun 9, 2014 at 23:58
• If exponentiation is not closed in Hamkins' system, then mustn't it be a partial function with certain demonstrable properties, i.e. some variation of the usual laws of exponents? Commented Jun 10, 2014 at 5:11
• You have a serious mix-up in the attributions here. It seems that you've confused JD Hamkins with Haim Gaifman. The position you describe and the quote you make appears to be from Haim Gaifman's paper, for which a link was made available on Hamkins's web site. Hamkins had provided a link to Gaifman's paper on his web site (to which you link) merely because Hamkins's seminar had read Gaifman's paper. The position you are describing and the quotation you make should be credited to Gaifman, not Hamkins. (I am Hamkins).
– JDH
Commented Aug 15, 2014 at 15:03