[FOM] From theorems of infinity to axioms of infinity

Alasdair Urquhart urquhart at cs.toronto.edu
Thu Mar 21 10:31:13 EDT 2013

Timothy Chow wrote:

> Similarly, Russell never doubted the axiom of infinity, but just had a strong 
> intuition that it was redundant to postulate it separately.  When this 
> intuition proved to be wrong, it should not be bewildering to find him 
> effectively shrugging his shoulders and saying, "Oh well, I guess we'll just 
> have to postulate it separately after all."

The content of the phrase "there are infinitely many objects"
changed dramatically for Russell between
1900 and 1910.  In the "Principles of Mathematics" the universe
of objects includes all conceivable things, concrete or abstract.
By the time of "Principia Mathematica", the notion of "individual"
is much more restricted, being confined to things that are not
logically complex.

So, the Axiom of Infinity in PM says (for the ground type) that
there are infinitely many individuals in this sense.
Whitehead and Russell do NOT postulate this as an axiom,
or "Primitive Proposition" in their terminology.
In fact, Russell in the "Introduction to Mathematical Philosophy"
goes so far as to say that the theorem of PM that there there
is at least one individual is a defect in logical purity.

The notorious proofs of the axiom of infinity by Bolzano and
Dedekind both involve implicitly the notion of the class of all thinkable
objects.  The rejection of the universe of all objects
as a mathematical object (due to the paradoxes) is the source of the 
difficulties of Russell and Zermelo; it leads them to postulate
the Axiom of Infinity (Zermelo), or write theorems
with the Axiom as an explicit antecedent (Whitehead and Russell).

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