Reduction of Mathematical Induction

[aleph_1]

To answer your first question, for first order theories, the model existence theorem (aka the completeness theorem) ensures that any consistent theory has a model. So, in that context, my answer to your question is 'no'. For other kinds of theories, I could not answer without specific context.

 

As to the intermediate value theorem, while some proofs may use choice, if I recall correctly, the intermediate value theorem is provable without choice (check me on that; my memory is not perfect and I don't have my notes with me now). 

 

As to the greater issue, of course I do recognize that choice is not constructive and leads even to such conundrums as that there is a well ordering of the reals but even in principle no specific well ordering of the reals can be defined.

 

As to axioms, I don't take them as statements that govern ALL mathematical situations, but rather I see a set of axioms as only governing those models in which the axioms are true. For example, the axioms for first order group theory are true for all groups but are false for many other mathematical situations. So too the axioms of ZFC are true for all models of ZFC but are false for many other mathematical situations. In other words, for me, an axiomatization is a "description" of a certain kind of mathematical context; so the axioms do govern in that context but don't govern in other contexts. 

 

In that regard, it doesn't make sense to ask me whether mathematical theorems are consistent with my philosophy. Mathematical theorems are not things I would even compare for consistency with my philosophical inclinations. For me, mathematics (putting aside for the moment consideration of applied mathematics, etc.) regards (1) the formal deductions themselves and (2) purely abstract structures. Neither of those are mediated by my philosophical inclinations.

 

As to construction of the natural numbers, again, neither the axiom of infinity nor choice need be involved. In set theory we may define the predicate 'natural number' and construct any particular natural number we wish without ever invoking infinity or choice. Indeed, using the von Neumann method, and even in greater generality, we can do the job with extensionality and three existence principles: that there exists an object, that for any object there is the set whose only member is that object, and that for any two sets there is the union of them. Finitistic and constructive.

GrandMinnow,

It seems that you are testing me. We are not talking about first-order theories and Godel's Incompleteness is operative here. The Intermediate Value Theorem depends on completeness. This is an independent axiom about which I do not currently find objectionable, philosophically. Your position concerning axioms reminds me of something I read by Novikov, the logicist. His position was that standard logic is valid when the axioms of logic are valid. When axioms such as LEM do not hold then standard logic does not apply. You claim that mathematical theorems are things you would not even check for consistency with your philosophical views. I recall Rand writing that your noblest act is the recognition that two plus two make four. I respond that some of what we call math is a recognition of reality and hence is pertinent to epistemology and philosophy more generally. Some math is useful for modeling and hence is provisional, subject to validation beyond mere proof. (Do you like the use of the word "mere" in this context?)

On my reading, the axiom of infinity is an essential element of the Von Neumann construction of the Natural numbers. However, I agree that there is a finitistic construction of the individual natural numbers.

What we may deduce from this thread is that the axiom of Infinity is essential to mathematical induction. Your acceptance of math induction depends on your acceptance of this axiom.

[GrandMinnow]

I still think that math induction requires the Infinity Axiom. This axiom asserts the existence of a collection of objects that is not finite. It is an actual set that is actually infinite. This set, the Natural Numbers, is required to have math induction.

 

Mathematical induction does not require having the set of natural numbers nor the axiom of infinity.

 

The following theorem schema is provable in Z/I (Z set theory without the axiom of infinity).

 

For any formula P:

 

(P[0] & Ax(x is a natural number -> (P[x] -> P[x+1]))) -> Ax(x is a natural number -> P[x])

 

is a theorem.

[GrandMinnow]

"It seems that you are testing me." [aleph_1]

 

I'm not. Indeed, you were asking me the series of questions.

 

"We are not talking about first-order theories" [aleph_1]

 

You asked me about consistency and models. I gave you the most exact answer I could regarding first order theories and said that I'd have to know the specifics of any other context. If there is some other context then you're welcome to specify it.

 

"Godel's Incompleteness is operative here" [aleph_1]

 

Yes, but what is your point in that regard? How does the incompleteness theorem bear on anything I've said?

 

"The Intermediate Value Theorem depends on completeness." [aleph_1]

 

Just to be clear, that is 'completeness' in a much difference sense than in either of two other senses mentioned: completeness (as in the model existence theorem) and incompleteness (as in Godel-Rosser). 

 

"This is an independent axiom" [aleph_1]

 

It's an axiom for some systems and it's a non-axiomatic theorem in certain other systems. 

 

"Your position concerning axioms reminds me of something I read by Novikov, the logicist. His position was that standard logic is valid when the axioms of logic are valid. When axioms such as LEM do not hold then standard logic does not apply." [aleph_1]

 

My point was not about any such provisional status of logical axioms. I was referring to the non-logical axioms that determine theories, which, if consistent, have various models.

 

"You claim that mathematical theorems are things you would not even check for consistency with your philosophical views. I recall Rand writing that your noblest act is the recognition that two plus two make four. I respond that some of what we call math is a recognition of reality and hence is pertinent to epistemology and philosophy more generally. Some math is useful for modeling and hence is provisional, subject to validation beyond mere proof." [aleph_1]

 

I don't claim that criteria beyond mere consistency are not of interest. For example, set theory is of special interest because it provides an axiomatization for the calculus, "the mathematics for the sciences". And of course we may be interested in comparing theories per the criterion of axiomatizing the calculus or many other criteria. But meanwhile, one may have intellectual or even just recreational curiosity in many consistent theories  (and even theories in alternative logics) that may or may not do well by various criteria. Then the only point I made is that I don't compare consistent theories for consistency with whatever philosophical views I may have. This does not restrict me from having particular interest in certain theories for various reasons and for comparison upon various criteria. Again, a consistent first order theory has a class of models. I take a model to be an abstraction concerning abstract relations; I don't hold that countenancing a consistent mathematical abstraction requires conformity to any philosophical views I may have. I don't intend to try to convince you to regard mathematical abstractions that way, nor do I hold my own "non-philosophical" view of mathematics as itself a philosophical thesis. Rather, you asked me a question, and to answer it honestly and meaningfully I'm informing you of my own approach.

 

"When axioms such as LEM do not hold then standard logic does not apply." [aleph_1]

 

Probably the most salient reason for working without the law of excluded middle is to ensure that all theorems are constructive in this sense: If I wish to prove merely that there exists an x having property P then classical logic is fine, but if I want to make sure that my approaches to such a proof also produce a particular object as an example of an x having property P, then I'd turn to intuitionistic logic (eschewing the law of excluded middle). 

 

"the axiom of infinity is an essential element of the Von Neumann construction of the Natural numbers." [aleph_1]

 

With the von Neumman method, the axiom of infinity is not needed to produce each natural number. What the axiom of infinity is needed for is to have a set that has all the natural numbers as members.

 

"the axiom of Infinity is essential to mathematical induction." [aleph_1]

 

The axiom of infinity is not needed merely to perform mathematical induction. The axiom of infinity is needed to have a set that has all the natural numbers as members. But mathematical induction itself does not depend on the existence of such a set.

Edited May 7, 2013 by GrandMinnow

[aleph_1]

GrandMinnow,

I appreciate that you keep me honest.

[GrandMinnow]

That is a very kind thing for you to say.