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What I am wondering is if mathematicians know whether (assuming consistency) the natural numbers are a definite object, without ambiguity. This seems intuitively obvious, but I don't know if its been shown. A formal question might be if two consistent axiomatic systems, each including all the axioms of the natural numbers and then some, can't contradict each other in statements about the natural numbers in the language of the natural numbers (not referring to objects guarunteed by the other axioms of the systems).

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  • $\begingroup$ Possible duplicate: math.stackexchange.com/questions/386550/… $\endgroup$
    – Asaf Karagila
    Jan 13, 2014 at 20:40
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    $\begingroup$ The question is not very clear to me. If you refer to formal number theory, like first-order Peano Arithmetics, it is known that this theory has different models (different structures th<t all satisfy the theory's axioms) that "looks different" (non-standard models vs the intended one). Speaking of "contradiction" about the "objects" is not correct, I think. A contradiction is obtained in a theory, when we derive from a set of axioms both the sentence $A$ and its negation ($\lnot A$). $\endgroup$ Jan 13, 2014 at 20:40
  • $\begingroup$ @ Asaf Karagila, they definitely have the same topic but my "formal question" does diverge from the other OP. IDK if they are duplicates myself. $\endgroup$ Jan 13, 2014 at 20:47
  • $\begingroup$ There is also an "effect" of Godel's Incompleteness Theorem which prove that in theories like first-order PA you can have undecidable sentences, i.e. a sentence $A$ such that nor $A$ nor its negation $\lnot A$ are provable from the axioms. In this case, you can add to the theory $PA$ in one case $A$ (call the new theory $PA_1$) and in the other case $\lnot A$ (call this one $PA_2$. Both are consistent, if $PA$ is, but of course they have different models. $\endgroup$ Jan 13, 2014 at 20:47
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    $\begingroup$ Bringing nonstandard models into the discussion is somewhat misguided, as they clearly do not represent the natural numbers, even if they agree on (some of) their first order properties. Some logicians speculate every now and then on the possibility of changing the theory of the standard model. Currently, no one seems to know how/whether it can be done. If that could be achieved, it would be a result much more significant that even the invention of forcing. It would show that there is inherent ambiguity, and would indicate the mathematical universe is much more subtle than currently believed. $\endgroup$ Jan 17, 2014 at 7:15

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The overwhelming consensus is that the true natural numbers is is a definite thing which objective properties -- not because this can be proved from anything more basic, but because mathematicians intuitively believe they have some kind of objective Platonic existence.

It is known, however, that not all properties of the Platonic naturals can be captured completely by any reasonable axiomatic system. Given any axiomatic system $T$ that proves no falsehoods about the integers, Gödel's incompleteness theorem shows how to construct an arithmetic sentence $\phi$ such that both $T\cup\{\phi\}$ and $T\cup\{\neg\phi\}$ are consistent theories.

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  • $\begingroup$ It would be better to say: given a recursively ennumerable axiomatic system $T$. Or even better, a recursively ennumerable theory $T$. $\endgroup$ Jan 17, 2014 at 18:33
  • $\begingroup$ Other than variations of the Liar Paradox (from Godel), have any serious limitations been discovered with the 2nd-order Peano axioms? $\endgroup$ Jan 18, 2014 at 5:09
  • $\begingroup$ The 2nd-order Peano axioms seem to me to perfectly capture the intuition of a single, never ending, non-branching, non-looping chain of nodes with some starting point. You couldn't weaken any of these axioms or remove any of them without admitting objects that shouldn't be. You couldn't strength them or add new axioms without eliminating objects that should be included. Have I overlooked any possibilities? $\endgroup$ Jan 18, 2014 at 5:47
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    $\begingroup$ @DanChristensen: Your strategy depends on quantifying over all possible predicates on the natural numbers, and what a "possible predicate" strikes me as being a lot murkier than what what a natural number is in the first place, so that definition doesn't really bring clarity. If you say that a predicate is just any subset, that assumes that you already know what sets and subsets are -- and it is indeed true that once you believe in set theory you get some $\Bbb N$ for free. But again set theory seems to require a larger leap of faith to start with than just numbers. $\endgroup$ Jan 18, 2014 at 18:51
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    $\begingroup$ @Dan Christensen: note that a nonstandard model of ZFC will believe that its natural numbers satisfy second-order PA characterization that you describe - so this characterization does not actually characterize the natural numbers until we have a way to characterize standard models of ZFC - which seems like a much more difficult problem. $\endgroup$ Jan 20, 2014 at 13:19
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A formal question might be if two consistent axiomatic systems, each including all the axioms of the natural numbers and then some, can't contradict each other in statements about the natural numbers in the language of the natural numbers (not referring to objects guarunteed by the other axioms of the systems).

If "all the axioms of the natural numbers" just means Peano Arithmetic, then it is completely possible to have two axiom systems that are each consistent, each contain all of Peano arithmetic, and are inconsistent with each other.

This is because there are many statements that are not provable nor disprovable in Peano arithmetic, even though the statements are in the language of Peano arithmetic. Therefore, if $A$ is such a statement, "Peano arithmetic plus $A$" and "Peano arithmetic plus the negation of $A$" are an example of the phenomenon I just mentioned.

One reason that many mathematicians think that the natural numbers are well defined (even if the powerset of the natural numbers is not) is that the natural numbers are all about finiteness, which we may think we have a good sense about. There are really three phenomena that are interrelated:

  • The natural numbers

  • The set of all finite strings on the alphabet $\{0,1\}$

  • The set of all formal proofs from an effective set of axioms

Any of these concepts can be interpreted, in a certain sense, in any of the others. For example, if we know what a finite proof is, we can interpret a natural number as the length of a proof. If we know what a natural number is, we can define proofs in terms of their Goedel numbers.

So, if the natural numbers are somehow vague, the concept of a finite sequence and the concept of a formal proof must be equally vague. Most mathematicians think we have some sort of perception of these objects, although a minority think the concepts are vague.

One of the most interesting recent developments in this respect is the potential in multiverse set theory (as developed by Joel David Hamkins and coworkers) there is no "standard" model of the natural numbers, and that every model of set theory is not well founded relative to another model. I do not think anyone has yet written a philosophical argument about the consequences of that for foundations of mathematics.

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    $\begingroup$ I just noticed this, Carl, and thanks for mentioning my work! As for further philosophical/mathematical consequences of the multiverse view, I would suggestion Francois's nice analysis at dorais.org/archives/1193. $\endgroup$
    – JDH
    Jan 31, 2014 at 2:19
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HINT: Given two Peano systems $(N,0,S)$ and $(N',0',S')$, prove there exists a bijection $f:N\to N'$ such that $f(0)=0'$ and $f(S(x))=S'(f(x))$.

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  • $\begingroup$ What do you mean by a "Peano system"? $\endgroup$
    – user642796
    Jan 14, 2014 at 6:33
  • $\begingroup$ $(N,0,S)$ is a Peano system iff $S$ is a successor function on $N$ that satisfies Peano's axioms. $\endgroup$ Jan 14, 2014 at 6:47
  • $\begingroup$ First- or second-order? $\endgroup$
    – user642796
    Jan 14, 2014 at 6:47
  • $\begingroup$ Second-order induction. $\endgroup$ Jan 14, 2014 at 6:52
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This answer is perhaps related to that of Carl Mummert. There is an informal but very inspiring article by Joel David Hamkins, about his caveats or doubts about if we are correct in thinking that we have an absolute concept of the finite. if Hamkins have doubts about it, I am pretty sure they should be taken seriously. All other number theorists that believe that the standard model of Peano arithmetic defines THE natural numbers beyond any doubts, should not be taken too seriously. You can read his inspiring thoughts at this link: http://jdh.hamkins.org/question-for-the-math-oracle/

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There is very little if any ambiguity in this case. For all practical purposes (putting aside Gödel), Peano's axioms (2nd order) can be used as The Definition of the set of natural numbers:

  1. $0\in N$

  2. $S: N\to N$

  3. $\forall x,y\in N:[S(x)=S(y)\implies x=y]$

  4. $\forall x \in N: S(x)\ne 0$

  5. $\forall P\subset N: [0\in P \land \forall x\in P: S(x)\in P \implies \forall x\in N: x\in P] $

It can be formally shown that if Peano's axioms hold on set $N$ with successor function $S$ and first element $0$ (as above), then the system $(N,S,0)$ is unique to within an isomorphism, i.e. all such systems are identical except for the names used. See: http://dcproof.com/EquivalentPeanoSystemsB.htm

EDIT: Here is a nice visualization based on the falling dominoes paradigm of the kind of "junk terms" (e.g. the disconnected side-loop shown) that is excluded by the induction axiom (5).

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    $\begingroup$ I don't see why you posted two answers. This is essentially an elaboration on the first answer, which wasn't that substantial (it's just a mere hint) to begin with. $\endgroup$
    – Asaf Karagila
    Jan 17, 2014 at 15:46
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    $\begingroup$ The difficulty with this, of course, is the second-order semantics are generally thought to be less clear and objective than the natural numbers. $\endgroup$ Jan 20, 2014 at 13:15
  • $\begingroup$ @CarlMummert Contrary to what GIT might tell you, it just seems to me that the above axioms perfectly captures the intuition of the natural numbers: a never-ending, non-branching, non-looping chain of nodes starting from a single point. Since all such formal structures are unique to within an isomorphism (i.e. essentially identical), these axioms might well be used as The Definition of the natural numbers. $\endgroup$ Jan 20, 2014 at 16:22

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