FOM: For Tragesser, Pratt, Franzen, et al.
Neil Tennant
neilt at mercutio.cohums.ohio-state.edu
Tue Mar 17 10:51:02 EST 1998
I have some questions, and some comments on earlier messages to
the list.
Questions:
1. Can anyone supply a reference for Ehrenfeucht's proof that all non-standard
models of Th(N) are elementary end-extensions of N?
2. Can anyone supply a reference to the first published proof that ZF
with Foundation, but with the axiom of infinity replaced by its
negation, has a non-well-founded model?
3. Does anyone know whether there is an English translation of the
THIRD, 1928 edition of Adolf Fraenkel's "Einleitung in die
Mengenlehre"? (The 1946 Dover edition is in the original German, by
the way.)
4. Has anyone seen any good movies lately? (Just kidding...Steve can
censor this bit if he likes.)
Comments:
1. For Robert Tragesser: you state Boolos's Conjecture as
"an intuitive [informal] proof that p is an intuition that p
is demonstrable in ZF" [or did Boolos mean ZFC?]
and you then mistakenly infer that
"if you begin with an informal proof and develop it into
a perfectly rigorous proof, one would end up in ZF."
I don't see why one *would* end up in ZF; no doubt one *could*, if one
decided to cast everything into set-theoretic terms; but why should
that be obligatory? It seems to me that Boolos's Conjecture can be
true without ZF-proofs being the inevitable termini of any and every
full rigorization of the original proof.
2. For Vaughn Pratt: you take issue (with Kanovei, I think) over
whether Dedekind defined his cuts as partitions into two sets A and B,
or in such a way that the cut could be identified just with A. For, on
the latter understanding, you point out that each cut would have its
"complement", hence not be unique. The original German was
Ist nun irgendeine Einteilung des Systems R in zwei Klassen
A1, A2 gegeben, ..., so wollen wir der K"urze halber eine
solche Einteilung einen Scnitt nennen und mit (A1,A2)
bezeichnen.
If I were translating this into English, I would have used "partition"
for "Einteilung", rather than "separation". It seems pretty clear to
this reader at least that Dedekind intended his conception of the cut
to be taken the way Kanovei suggested. Dedekind goes on to say that
each rational number produces two cuts, depending on whether it is a
member of the left set or a member of the right set. But nowhere (so
it seems) can we impose the construal that he *identified* the cut
with the set A1 rather than with the *partition* involving both A1 and
A2 symmetrically.
3. For Torkel Franzen: You suggest that the importance or general
intellectual interest of Friedman's result depends on
"the applicability of the combinatorial principles and the
epistemological status of the large cardinal principles."
I disagree with this. The importance, for me, of the independent
statement being a combinatorial one is simply that that makes it more
intuitively graspable as something that darn well *ought* to have a
truth-value. Whether it has applications seems to me
irrelevant. Furthermore, the *shakier* the epistemological status of
the large cardinal principles, the more arresting the
result!---because Friedman has shown that this relatively simple
combinatorial statement is no more secure than the large cardinal
principle. His result *puts on an epistemological par* the two kinds
of principle---combinatorial and large-cardinal. *That* is why it's so
impressive. If I am told that someone has shown "A iff B", when that
sort of equivalence would have struck me, prima facie, as highly
unlikely, I don't question the significance of the proof of that
equivalence by saying "Well, we'll have to wait and see whether A has
any application; and we'll have to wait until we've settled the
epistemological status of B." If A looks like the kind of claim that
ought to be true for simple reasons, and B is the kind of claim of
which I already know that many people claim not to know how one might
come to know it, then I'm willing to state that I'm impressed.
Neil Tennant
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