Re: Logarithm of transfinite numbers

RLG wrote:
"Robert Low" <mtx014@xxxxxxxxxxxxxx> wrote in message
news:48khiqFkpho9U1@xxxxxxxxxxxxxxxxx
RLG wrote:
First order PA is rock solid, Tony has no grounds to question
that. Second order PA is more up in the air; many people, like
Quine for instance, have raised legitimate objections to second
order logic.

Well, if you're concerned that the von Neuman model
isn't adequately characterised by the second order version
of PA, then you'll stay that way, I guess. Most of the
rest of us find the argument fairly convincing.

It is a philosophical argument either way, since the problems with
second order logic are well known among logicians. In this respect,
second order PA is like the axiom of choice or Godel's axiom of
construction. One can find 'convincing' arguments for and against
each of them.

Set theory, of course, has philosophical ramificaations, but set theory
itself is not philosophy. Seeing what follows from what axioms, what
models are models of what theories, and looking at theorems as to
comparisions among theories and models. Those are all mathematical
actitivities. And you are incorrect that the embroilment with Tony
amounts to a mere subjective difference in preference for one theory
over another. Rather, the problem is getting Tony to see the difference
between a mathematical theory, with axioms, definitions, and theorems
and, on the other hand, metaphorical musings in a private,
non-formalized mathematical language without axioms and definitions, NO
MATTER WHAT AXIOMS are in play as regards the theorems. Then, too
often also, is the problem of getting Tony to understand that something
is or is not a theorem of a particular theory, which, in context, is
almost always PA or Z or variants.

MoeBlee

.

Relevant Pages

  • Re: Set Theory: Should you believe?
    ... "second order theory", i.e., one which is couched in a formal language ... a set of first order axioms asserting that any first order definable subset ... NOT "confused" for you to intuit one thing and me to intuit another ... IN FIRST-order PA and if ANYbody fails to bother, ...
    (sci.logic)
  • Re: Second order arithmetic and higher order arithmetic
    ... There are canonical axioms for full second order arithmetic ... ZF set theory. ... most mathematics of interest to most mathematicians takes place in the ...
    (sci.logic)
  • Re: What is the 1st order formal system known as PA?
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    (sci.logic)
  • Re: Very easy problem but I am a stupid non-logician idiot
    ... wikipedia in case you think I mean something else). ... clearly induction has to be used somewhere to define addition. ... the system of second order wikipedia arithmetic doesn't ... mention anything about + or x or> in the axioms or language. ...
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  • Re: Turing completeness of the functional paradigm?
    ... sets of axioms which uniquely define the natural numbers: the second order Peano axioms do this. ... and second-order languages ... the (second order) PA axioms were categoric, which meant that there was essentially only one model. ... is the whole of N) not to give uniqueness except that ...
    (sci.logic)