Re: Anti-diagonalist page

Capitals: on my printout of the anti-diagonalist page, the name Ulrich
ends a passage on which there are capitals. I now find that the
printout does not include Ullrich's name at the start of his messages,
for some reason - other people's names in that place are printed out
well enough.

1) Does 1-1 correlation equate with equal numerosity? The even numbers
can certainly be 1-1 correlated with the odd numbers, but a further
premise is required to derive from this that they have the same number,
namely that either of them has a number. Cannot one simply define
'power', so that having the same power is equivalent to being 1-1
correlatable? One still needs an existence proof that the function
'the power of the Ps' is not partial. This point is made by Boolos in
his book 'Logic, Logic, 'Logic', about his function '#', and the
general question of whether abstraction principles can be just
stipulated has the been the subject of a long debate in the
Neo-Logicist tradition - see, for instance, most recently, 'What is
Wrong with Abstraction?' by Michael Potter and Peter Sullivan,
Philosophia Mathematica (III) 13 (2205) 187-193.

2) 'I gather that you're not one of [the people who insist that the
proof that there is no bijection between the reals and the natural
numbers is wrong]'. If the whole of my paper 'The Uniform Solution',
and particularly the argument at the end, had been read, as I have
suggested, my position would not have needed to be just 'gathered'.
Certainly there is no bijection, but that does not entail what Ulrich
takes it to entail - as the paper by Wright, previously referenced, in
an even fuller way has shown. Wright's point, about the wider
implications of Skolem's Paradox, is more briefly made in my paper,
since in my paper it is just an illustration of a more general point
about a large number of paradoxes which arose in logical theory at the
end of the nineteenth century.

3) The meaning of the word 'axiom' has changed since Euclid, says
Ullrich. Indeed, an 'axiom' these days is no longer taken to be a
self-evident truth, but if its truth is not in question then all one is
left with is the sentence, independent of any interpretation. So is it
then the case that 'the existence of an infinite set is an _explicit_
axiom of set theory'? Certainly there are sentences in Set Theory
which are taken to have that interpretation, but the mathematics
proceeds independently of any interpretation. If you want to say some
axiom states certain infinite sets exist then you are involved with an
interpretation, not the mathematics. Skolem's paradox, as above, shows
that there is an interpretation of Set Theory in which there are no
non-denumerable sets; the work of Lavine, following Mycielski, shows
there is even a finitary interpretation of Set Theory - see
'Understanding the Infinite' (Harvard Univesity Press, Cambridge MA
1994).

.

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