Re: Question about induction

Aatu Koskensilta <aatu.koskensilta@xxxxxx> wrote:
Bill Dubuque <wgd@xxxxxxxxxxxxxxxxxxxx> writes:

I disagree. It's pedantry only to mathematical logic cognoscenti.
For others it is a point that does deserve some emphasis. From my
experience here it is clear to me that the leap from w to higher
ordinals is not something that necessarily comes intuitively.

What prompted my (rather benign) charge of "logical pedantry" was your
description of the claim that least number principle and the principle
of mathematical induction are equivalent as "logically naive" and "false".

But it almost always is when made by most logical laypersons, including
mathematicians who have not studied mathematical logic -- see below.

When we assert that two mathematical statements are equivalent we mean
that one can be mathematically proved from the other. Usually such
assertions correspond to a logical result, that the formalisations of the
statements are provably equivalent over some appropriate base theory.

But I already said precisely that (and more) in [1] (which I linked to).

But what is the appropriate base theory? We must use our good
sense and best judgment to figure that out, and as you note, in this
instance it's not Peano arithmetic with the induction axiom (schema)
dropped, but rather the theory incorporating the basic properties of the
successor function on naturals. This is a piece of logical arcana of no
obvious mathematical significance, and certainly does not mean that the
claim that the principle of mathematical induction and the least number
principle are equivalent is false.

Since you seem to be repeating what I said, I can only presume that you
didn't follow the link [1] in my original post (and the links there...)
If you do you will find similar logically sloppy/naive/erroneous posts
made by many respected sci.math folks (e.g. Chapman, Magidin, Pratt...)
as well as published papers that do likewise (and AMS reviews thereof).

Indeed, but the OP (and most other armchair logicians) don't realize
that they also need to explicitly include that axiom (or another axiom
that suffices to characterize w).

The axiom that all non-zero naturals are successors does not characterise
omega.

Did somebody claim that it did?

E.g. one does not obtain an equivalent form of Peano's axioms simply
by replacing induction by well-ordering or the least number principle
- as many logical laypeople naively seem to think

I have no reason to doubt your experience in this matter. Such logical
misconceptions certainly merit correction.

Again, if you read the linked posts perhaps you may better appreciate
the motivation for my objections. I think perhaps you are assuming far
too much of the logical layperson. The truth may surprise you. As the
saying says: a little bit of knowledge can be a dangerous thing ...

--Bill Dubuque

[1] sci.math, 23 Feb 2005, Can transfinite (strong) induction
for N be derived from Peano's Axioms?
http://google.com/group/sci.math/msg/0f55d423504808b1
http://google.com/groups?selm=y8zvf8i3cg5.fsf%40nestle.csail.mit.edu
.

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