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In my career I've been formed mostly in the formal side of mathematics, that is, standard set theory and every classical branch of mathematics that uses set theory.
However, I am not quite sure about the common "rules" that are accepted in meta-math and, more specifically, in the foundations of mathematics.

I have several questions, but all of them are related to the nature of intuitive natural numbers.
I will start from the formal side, to explain my point, but I am really interested in the intuitive and metamathematical issues.

From the formal point of view, we have a "set" of natural numbers, $\mathbb N$, which can be obtained for example by the Von Neumann construction:

$$0 = \emptyset$$ $$1 = \{0\}$$ $$2 = \{0,1\}$$

and so on.

The existence of a set containing all these "formalized" natural numbers comes with the development of set theory. To be concrete, I will talk about ZFC.

However, ZFC has to be sustained in the first order logic, and both are formally expressed in the "language" of first order logic, which consists of a collection of symbols $\mathcal L = \{\wedge,\vee, \forall,\exists, ...\}$, and rules to form formulas with that symbols.

FIRST ISSUE. It is common to see in the books of logic that the collection $\mathcal L$ of symbols of the language is "finite". Moreover, the word "set" is used (I am using "collection" in order to not merge things of different contexts). The meaning of "finite" in the metamath context cannot be referring to any notion of "finite set" of ZFC, because this would be circular (the problem is that, if I am "constructing" set theory from "nothing", and the notion of "finite set" comes later, after some theory is done, I cannot talk about "finiteness" in the "ZFC sense"). So:

FIRST QUESTION. In what sense is it understood that the collection of symbols $\mathcal L$ (intended to construct first order logic) is considered "finite"? In what sense is it understood that the length of a formula of first order logic is finite? Is it implied by a previously accepted notion of intuitive natural number? Please, I need to know the standard point of view, and not opinions of a very personal kind.

SECOUND ISSUE. When methamath theorems are demonstrated, sometimes some "properties" of natural numbers are used, as for example the induction on the number of symbols in a given logical formula. Again, these properties are referring to the intuitive natural numbers.

SECOUND QUESTION. How can I be sure about what properties of natural numbers can be accepted or not, in the metamath context? Is there some kind of consensus about what intuitive properties of natural numbers can be used?

THIRD ISSUE. If a formal first order theory contains the Peano axioms, then on the semantic side there are a lot of non-standard models... However I am even more intrigued by the "standard model", which is, again, the intuitive natural numbers.

THIRD QUESTION. Are they actually a "model"? In order to prove that, what properties of intuitive natural numbers are commonly accepted as "true"?

FINAL QUESTION. Does there exist an agreement about what the intuitive natural numbers are and which of their properties are accepted and used in metamaths?

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For formal number theory, we need not worry about what finite may mean, since the list of function symbols, constant symbols is explicit (and very short). –  André Nicolas Jun 20 at 15:01
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(Opinion): Mathematical logic, model theory are branches of mathematics, just like analytic number theory, or numerical analysis. The methods, ideas used in the first two branches mentioned need no greater (and no less) scrutiny than the ones in the other two. –  André Nicolas Jun 20 at 15:15
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Define intuitive natural number. –  William Hilbert Jun 27 at 17:22
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@William Hilbert: If your asking me for a definition of intuitive natural number, it means that you are missing the point. The upvotes you obtained in your comment are wrong, also. –  pablo1977 Jun 29 at 19:41
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@OlivierBégassat, let me know if you can count up to the number of elementary particles in the universe (plus 1). –  user72694 Jul 31 at 12:54

4 Answers 4

The usual answer (or dodge, depending on yuor philosophical position) is to talk about the intended model (more precisely, intended interpretation) of the natural numbers. If such a thing exists (and many mathematicians do believe it exists, including the constructivist Errett Bishop), then one can interpret references to "finite" at the meta-level as referring to things equinumerable with individuals in the intended model. If you don't want to believe in an intended model then you also have to give up hope of an absolutely rigorous development of mathematics "from scratch". Whether or not you actually lose anything in the process again depends on your philosophical position. Kronecker, by the way, never expressed any opinion on the matter in writing. Whatever is reported in his name is hearsay based on Weber (who certainly made a mistake when he mentioned "whole numbers" rather than "natural numbers").

Juat out of curiosity, I looked up "intended interpretation" on wiki, and was led to the following comment:

Intended interpretations. Many formal languages are associated with a particular interpretation that is used to motivate them. For example, the first-order signature for set theory includes only one binary relation, ∈, which is intended to represent set membership, and the domain of discourse in a first-order theory of the natural numbers is intended to be the set of natural numbers. The intended interpretation is called the standard model (a term introduced by Abraham Robinson in 1960).[8]

Note the definite article, which I guess begs the question (namely, yours). Here the reference is to the paper

Roland Müller (2009). "The Notion of a Model". In Anthonie Meijers. Philosophy of technology and engineering sciences. Handbook of the Philosophy of Science 9. Elsevier. ISBN 978-0-444-51667-1

which you may find useful (though I hasten to admit that I never read it). If you gain some insights do let me know.

The book that the article is in can be found here.

For a related question see What are natural numbers? where you will also find an accepted answer containing a passionate defense of the intended interpretation without mentioning the term ("categorical" and all).

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+1: Thanks. At least you understand what about I am talking about. –  pablo1977 Jul 31 at 11:03

I will start by saying that I don't think there is exactly a "standard opinion" for all of these questions so I will give you what I have gathered from my logic professor and my readings.

I think it is important to understand that we assume ZFC axioms in order to do logic. Thus we already have definitions of finite when we start to define our first order logic. Likewise we can prove, from ZFC, that induction works on formulas works. I think this should hopefully clear up question 1 and 2.

For issue 3 I would like to note that the only reason that we have non-standard models is that the Peano axioms (specifically induction) are not fully expressible in the language of first order logic. Our best attempt at a first order formulation of the axioms is what has non-standard models.

For question three we have a similar situation to one and two. We are working within set theory when we do logic so the set of Von Neumann natural numbers does exist and is a very real model.

While I have addressed the issues regarding logic the question of whether there is a "standard intuitive notion of natural number" remains open. There is really no way for mathematicians to say for sure that they all have the same notion of natural number.

That being said, I think that almost every mathematician believes we share the same notion of $\mathbb{N}$ (Likewise for reals and rationals and other things). This belief is mostly due to the fact that, as far as I know, there are not any properties of natural numbers that people disagree on.

This is not to say that there is an intuitive notion for all mathematical objects. When it comes to sets, some people still do not think that they have the choice property. Thus there are multiple intuitive notions of set.

Edit: Judging from the comments I need to give concrete evidence of the use of ZFC in logic proofs. The only one that I can think of off the top of my head is in my advisor’s paper on Dedekind-finite sets http://www.math.lsa.umich.edu/~ablass/ddiv.pdf Proposition 2.1 is a proposition about permutation models which explicitly uses the axiom of choice.

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I think it is important to understand that we assume ZFC axioms in order to do logic. This is simply wrong. You are not well informed. Godel Completeness Theorem doesn't need any ZFC at all, and it is a Theorem about 1st order logic. –  pablo1977 Jul 15 at 16:39
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Godel Completeness Theorem states the existence of a specific model. A model is a set. How exactly do you suppose we can prove the existence of this specific set if we do not have any axioms governing what a set is? Another way to put it. You can not prove a statement without making any assumptions. So what are the assumptions you are taking when doing logic? –  Dylan Stephano-Shachter Jul 15 at 17:17
    
The problem is that you are missunderstanding the use of "set" in model theory. A "set" in model theory is not a ZFC set, but a naive-set-theoretic set. This is, again, of intuitive nature. The user tomasz is being the right approach to the topic. Please read his comments and check if you properly understand what we are talking about. –  pablo1977 Jul 15 at 17:37
    
I am fairly certain you are wrong about this. The whole reason mathematicians believe the results of logic is that it is rigorous. You can not base rigorous results on a naive theory. Look through any logic book you want, I can assure you they will not use anything that contradicts ZFC(Well at least ZF). –  Dylan Stephano-Shachter Jul 15 at 17:56
    
Let me explain it in one more way. We use sets and their (possibly naive) properties to prove theorems in logic. So we end up essentially proving "[some statement] is true assuming that sets work the way we think". This, however, is a completely ambiguous sentence. The only way we can make it unambiguous is to define "sets work the way we think". This is where axioms come in. –  Dylan Stephano-Shachter Jul 15 at 18:05

This question, besides intriguing, has been a waste of comments, answers and bounties. Now that it's all over, allow me to throw in my 2 cents worth and enlighten a few more issues, in an attempt to address the OP's issues and questions (again, second attempt), though with a twist.

The title alone is problematic already. Does there indeed exist a standard role of intuitive numbers in the foundations of mathematics ? If the answer would have been yes, then certainly there would have been sort of consensus by several experts at the MSE forum here. But all of the comments and answers indicate that there is no such role . Therefore let's cut the title in two pieces, in order to get two more amenable questions:

  • the standard role of numbers in the foundations of mathematics
  • the role of intuitive numbers in the foundations of mathematics
The first version of the title, most probably, would have given rise to the Peano axioms, or the von Neumann construction; all very standard. But the second version of the title gives rise tot a non-standard answer, though not an "opinion of a very personal kind" either, which is my next point.

Intuitive natural numbers are defined as the very first thing in Intuitionism . Quoted from this reference: The existence of the natural numbers is given by the first act of intuitionism, that is by the perception of the movement of time [ ... ]. Therefore intuitionism seems to be fine in its a priori conception-without-axioms of the naturals, which hence may be truly called intuitive numbers . However, the lack of the excluded third rule in intuionism seems to be a major hurdle for the OP and is judged as nonsense by the major part of mathematicians nowadays, as said below. Indeed: intuitionism is not standard. But I think it does no harm if someone takes notice of the very first principles of intuitionism - and eventually forget about the rest. (Inasmuch as reading the first few pages from "Das Kapital" doesn't make someone a communist!) In a nutshell, we are left then with intuitionism-without-the-trouble ("pink" intuitionism) / sort of naive constructivism as uttered by Leopold Kronecker : Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk (rendered in English as: God made natural numbers; all else is the work of man). So these must be the OP's intuitive numbers. I can't think of anything better.

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Hilbert and Godel did not accept the framework of intuitionism, but they accepted the existence of natural numbers in an intuitive manner. So, I think that this answer is not adequate. Besides, you say that "it is inevtiable that you start with accepting the framework of intuitionism/constructivism...". To just "accept" things is act of faith, it is not science. Why do I have to accept it? What is implied in that "acceptance"? The lack of 3rd exluded rule in intuionism it seems as having nonsense by the major part of mathematicians nowadays. –  pablo1977 Jun 24 at 1:39
    
@pablo1977: It seems to me like your question is not only very vague, but also impossible to answer for the reason that you seem to be looking for a very specific kind of answer which just can't be given in this case. You may consider accepting a convention an act of faith, but this act of faith is necessary for any science, and indeed, any activity. We need some rudimentary assumptions, some primitives with no definition which we simply accept. Without that we're suspended in limbo. That said, I'm far from being a proponent of intuitionism in mathematics. ;) –  tomasz Jul 18 at 11:10
    
@pablo1977: Therefore, I'm voting to close. –  tomasz Jul 18 at 11:10
    
My question is not about whether the intuition is valid or not, but what are the rules accepted for intuitive natural numbers. I can accept primitive intuition as part of foundations, but the problem is that people is vague when they have to say what rules, properties or theorems are in a the "intuitive" domain and which are not. My question is precise, and I am looking for precisions that people in general don't take care enough to point out in the books. –  pablo1977 Jul 18 at 11:21
    
About what your answer: you put in doubt if the intuitive natural numbers have a rol in foundations. Yes they are, because in the foundations we have to build-up first order logic and set theory from "nothing". The Peano axioms or other constructions in set theory are becoming very "later". But to proof some metatheorems, natural numbers and recursion are used in several ways, the qualifier "finite" (in the sense of "counting up to some n") is used everywhere referring to the length of logical formulas, and so on. So, a primitive or intuitive use of natural numbers is everywhere. –  pablo1977 Jul 18 at 11:31

For the first question, it depends on the context. When talking about strict foundations, finite would usually mean finite in the intuitive sense. This is rather important in this context – firstly it allows us to avoid (or postpone) ontological and intuitive pitfalls related to infinity, and secondly it allows us to encode everything using some Gödel numbering.

For second question, I believe Peano arithmetic is generally accepted, and sufficient for the proof of most basic foundational facts.

For the third question, I don't think there is much controversy about what a natural number is. A natural number is an object which can be obtained by adding $1$ to itself some finite number of times (though “finite” may be controversial). If the set of all natural numbers exists, it is certainly a model of any interesting theory of arithmetic: they are all approximations of what the natural numbers are like.

The last question is covered in the previous one, I think.

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I am not satisified. The problem here is that "finite" can be defined through the intuition of natural number, and natural number can be defined from the concept of "finite". The "controversial" point remains being essentially the same. If there are "controversial" issues in the "foundations", then mathematics is not a reliable science. A solid starting point is need. About the 2nd question, is part of foundations too, because there the properties of N are used to index variables in 1st order logic, or to give proofs by induction in the length of formulas. –  pablo1977 Jul 15 at 16:48
    
@pablo1977: It is not reliable in the sense that you can't rigorously show that the foundations are consistent (although the same is true for all sciences, I guess). Most mathematicians believe that they are, but this is a problem with all kinds of foundations (not just mathematical): you must take some primitives for granted (or risk circularity or infinite regress). For the most part, foundations can be reduced to basic (naïve) arithmetic or naïve set theory, and there's no solid starting point for those beyond our intuition coming from some real-world experience. Mathematics is not science. –  tomasz Jul 15 at 17:02
    
Agree with you in all, except by the last phrase. Mathematics is science, or part of the science. It needs rigor, because is the language of a lot of important scientifical brances. Anyway, going back to the intuition of natural numbers, you have put the things in the right way: the naive natural number theory is a primitive which mathematicians believe true. So, I can here restart my question: what is exactly the form and properties of this "naive" theory of natural numbers? When I use a property of N in a "foundations" paper, which criteria will use the referee to say can/cannot be used. –  pablo1977 Jul 15 at 17:22
    
@pablo1977: My point was exactly that there are no exact criteria, just like there are no exact criteria as to what is and isn't a proof. As to whether or not mathematics is science, many would not agree with you. See for example this thread: math.stackexchange.com/q/287701/30222 . –  tomasz Jul 15 at 17:30
    
OK, but we have "foundations" magazines, with papers there. The referees have to judge with some criteria. If I use a naive-N property to proof something in metatheory, how the referee knows if this particular step is right or not? –  pablo1977 Jul 15 at 17:35

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