[FOM] Axioms of reducibility and infinity

Alasdair Urquhart urquhart at cs.toronto.edu
Sun Aug 7 13:59:14 EDT 2011

Wittgenstein's reasons for rejecting the axiom of infinity
are quite clear.  As stated by Whitehead and Russell
in Principia Mathematica, it says that there are infinitely
many individuals (i.e. objects of the lowest type).
Clearly there is no reason to think this is true a priori
of the world (the Tractatus is an attempt to describe
the a priori logical structure of the world).
In other words, in the construal of Whitehead, Russell
and the early Wittgenstein, the Axiom of Infinity
is an empirical postulate -- there is no reason to think
it is a logical truth.

I have never understood Wittgenstein's reasons for
rejecting the Axiom of Reducibility, and always found
his discussions of it quite obscure.

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?
> Francisco

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