[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 
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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