[FOM] Axioms of reducibility and infinity

steve newberry stevnewb at att.net
Mon Aug 8 17:06:32 EDT 2011


My understanding of the Axiom of Reducibility is that it was intended to state that:

To every proposition of higher-order, there is an equivalent proposition of First-order,
or more precisely, to every entity definable in Higher-order logic there is an equivalent
such entity definable in  First-0rder logic.

If AXIOMATICALLY true, then there is no ontological difference between First- and
Higher- order logic, which is now well known to be untrue, and Wittgenstein may
well have intuited that fact.

Nicht wahr?


Steve Newberry

--- On Sun, 8/7/11, Alasdair Urquhart <urquhart at cs.toronto.edu> wrote:

From: Alasdair Urquhart <urquhart at cs.toronto.edu>
Subject: Re: [FOM] Axioms of reducibility and infinity
To: "Foundations of Mathematics" <fom at cs.nyu.edu>
Date: Sunday, August 7, 2011, 10:59 AM

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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