[FOM] Solution to Buckner?
Dean Buckner
Dean.Buckner at btopenworld.com
Wed May 7 13:05:51 EDT 2003
Harvey,
Thanks for the note. But I am a little confused about what constitutes PA,
PA' and ACA0.
I believe natural language embeds some consistent set of assumptions or
"axioms" sufficient for dealing with pluralities. I am interested in
seeing whether such assumptions are consistent with (preferably support) any
"ordinary mathematics". The most important constraint embedded in natural
language is (I believe) that there are no infinite sets, i.e. objects to
which infinitely many objects bear the "membership relation". I.e. it
embeds an Axiom of Finity, if you like.
I always understood that PA not only excluded the Axiom of Infinity, but
included its negation. This suggests it is a natural analogue of "NL" - the
system embedded in natural language.
But do PA' and ACA0 therefore include the Axiom of Infinity? If they do, I
can't see this from the definitions you give. If not, a cursory inspection
of Prof. Simpson's book (Chapter 1 on website) suggests that ACA0 supports
the concept of real numbers, how is this consistent with not including the
Axiom?
Concerning Cantor's Theorem, there are two versions of this. The version
so-called by Zermelo is that "every set is of lower cardinality than the set
of its subsets". NL would contain this as a Theorem. The version in almost
every popular version I have read (and that is now quite a few) has it as
proving "the uncountability of the real numbers". This requires two further
assumptions
(i) some sets are infinite
(ii) without infinite sets, some real numbers would not exist
Common sense, and the evidence of popular works on the subject suggests that
(ii) is correct. For example it's commonly said that the Axiom of Infinity
or some version thereof is required before we can get to work on real
numbers. Moreover, suppose (ii) were false, i.e. that the existence of any
real number could be demonstrated without requiring infinite sets. Then,
Lo! you could easily match up every natural number with some real number, so
long as you had infinitely many natural numbers to use for matching. But
this would suggest there was an easy solution to the continuum hypothesis.
But there is NO easy solution to the continuum hypothesis, therefore (ii)
cannot be false. Anyway, common sense suggests that, if there are
infinitely many objects of a certain kind, there must be a set of them.
So, the evidence suggests that real analysis requires the Axiom of Infinity,
or some variant, and that therefore NL (the set of asusmptions embedded in
ordinary language) does not support real analysis.
A further point: in your posting you say an option would be to "Rip apart
and reassemble real analysis in order to do it in PA". But if the common
sense view is true, and if PA does not embed the Axiom of Infinity, this
wold surely be impossible. Are you seriously saying that it is not
conclusively established that PA cannot support real analysis?
Forgive these questions from someone for whom mathematics is an "area of
incompetence" - and many thanks for the time you have spent in this small
corner of the foundations.
Kind regards,
Dean
Dean Buckner
London
ENGLAND
Work 020 7676 1750
Home 020 8788 4273
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