[FOM] RE: FOM New Axioms
Matt Insall
montez at fidnet.com
Fri Jun 27 11:15:07 EDT 2003
[Todd Eisworth]
What
does the Axiom of Choice possess that the Continuum Hypothesis does not?
[Matt Insall]
Consider the following statements:
a(X) The set X can be well-ordered.
b(X) There is a subset Y of P(X) for which there is
no bijection from X onto Y and there is no bijection
from Y onto P(X).
Then the following are provable in a very weak fragment of ZF
(Please do not ask me how weak. I just know that not very much
of ZF is needed to prove either of these.):
(i) For every finite set X, a(X).
(ii) For every sufficiently large finite set X, b(X).
Thus these are (finite) combinatorial principles that are true
without appeal to strong axioms. I happen to think that the
intuition that gave us the axioms of ZF was, or at least can be,
grounded in the following principle:
(**) When possible, sets behave as do finite sets. Otherwise,
proofs or new axioms are required to explain why some sets
stray from behaving as if they aere finite.
Thus, when a statement is consistent with previously chosen axioms
and has a ``relativization'' to finite sets that is provable from those
axioms, we should choose that statement to hold for all sets,a s a new
axiom.
The following are consistent with ZF:
(p) For every set X, a(X).
(q) For every sufficiently large set X, b(X).
Statement (i) is the axiom of choice, and statement (ii) is the
negation of ths generalized continuum hypothesis. But we can go
further than (ii). This is because there are other statements that
imply (ii) that are not decided by ZFC+~GCH. We have the following:
(ii)' For every finite set X with more than one member, b(X).
Thus, the principle (**) suggests we adopt as an axiom
(q)' For every set with more than one member, b(X).
In the theory ZFC+(q)', ~CH is provable, as is ~GCH, but in fact,
this goes much further. It imposes on every infinite cardinality
\kappa the inclusion of a cardinal number between \kappa and 2^{\kappa}.
Moreover, these axioms should not bother finitists, for if one casts
out the axiom of infinity, or negates it, then these axioms make no
mention of the infinite, so they still make sense in the resulting
theory. (Indeed, in axiom systems that entail that every set is finite,
these should be theorems.)
We can go further. Consider the following:
(iii) As n approaches infinity, 2^n-n approaches 2^n.
This is statement of analysis about finite sets. An extension of (**) to
include such asymptotic results in the world of finite sets would be
(***) Infinite sets behave as much like finite sets as possible, in an
asymptotic sense.
Applying this principle to the problem of CH and the problem of GCH yields
very strong negations of both. In particular, we get
(r) For any infinite cardinal number \kappa, if X is the set of cardinal
numbers
strictly between \kappa and 2^{\kappa}, then X has cardinality 2^{\kappa}.
This is a way to settle GCH (and CH) that has as its philosophical basis the
analysis of finite cardinals. In fact, the principle (***) can be applied
in place of (**), to make the principle closer to analysis, which is the
original
home of Cantorian set theory. Nevermind that Cantor and his contemporaries
did
not apply (***) when proposing CH! In this sense, it seems to me that they
changed the way they were applying their analysis intuition to develop the
theory of sets.
By the way, I understand that my strong negation of (G)CH is not the one
supported by current arguments from analysts and topologists. I still need
to learn more about why these analysts and toppologists prefer statements
like
2^{\aleph_0} = \aleph_2,
because at this point, this axiom seems to me to be an ad hoc one that has
no
clear principle like (***) behind it.
Matt Insall
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