[FOM] Axioms of reducibility and infinity
William Tait
williamtait at mac.com
Mon Aug 8 11:19:26 EDT 2011
On Sun, 7 Aug 2011, Francisco Gomes Martins wrote:
> I´m working on Tractatus; Wittgenstein rejects
the axiom of reducibility (see
<tel:%286.1232-6.1233>6<tel:%286.1232-6.1233>.1232-6.1233),
the axiom of infinity (5.535) and even the even
the set theory (6.031). First, I ´d like to know
more about those axioms. Second, I´d like to know
why/how does Wittgenstein reject all of them?
I am very sorry to hear that you are working on
the Tractratus, since I fear that it may well
drive you mad. But here are two things that I
remember, before I escaped (I think), that may
answer your first two questions. The first is
that somewhere is the statement that the axiom of
infinity is expressed by having an infinite
number of individual constants in the
language---or it may be that it is expressed by
having a name for each natural number. (For the
reason that I have already given, I am reluctant
to go back and find out which of these it
is.) The second is that every proposition is a
truth function of elementary propositions. Since
W is contemplating that there are an infinite
number of individual constants, presumably he is
contemplating infinitary truth functions, since
"for all individuals x A(x)" must be "A(a) and
A(b) and A(c) and ...", where a, b, c, ... are the individuals.
So, I am not sure why one would say that he rejected the axiom of infinity.
It IS clear why he would reject the axiom of
reducibility, however. Lets just think of
propositions of ramified second-order. The
proposition "for all X^a B(X)" where X^a ranged
over the sets of individuals of rank a, can be
regarded as the infinite conjunction "B(T_0) and
B(T_1) and ...", since the second-order
quantifiers in the second-order terms T_i =
lambda x B_i(x) are all all of rank < a; and so
it follows that the propositions of ramified
second-order can indeed be regarded as
truth-functions of the elementary propositions.
But the axiom of reducibility amounts to
eliminating rank and so a proposition "for all X
B(X)" means "B(T_0) and B(T_1) and ..." where the
"T_i" now range over ALL second-order terms "T_1 = lambda x B_i(x)". Now let
B(X) be "for all X (0 \in X)" where "0" denotes
some individual, and let T = \lambda x for all X
(x \in X). Then one of the conjuncts B(T) of "for
all X B(X)" is itself: B(T) = for all X B(X). So
the method of eliminating second-order
quantifiers by means of infinitary conjunctions
(and disjunctions) encounters circles and so does
not work in the case of impredicative logic. So
the axiom of reducibility, it is not clear how
one would justify the thesis that all
propositions are truth-functions of elementary propositions.
Bill Tait
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