Full Idea
There is something profoundly unsatisfactory about the axiom of infinity. It cannot be described as a truth of logic in any reasonable use of that phrase, and so the introduction of it as a primitive proposition amounts to the abandonment of logicism.
Gist of Idea
The axiom of infinity is not a truth of logic, and its adoption is an abandonment of logicism
Source
W Kneale / M Kneale (The Development of Logic [1962], XI.2)
Book Reference
Kneale,William and Martha: 'The Development of Logic' [OUP 1984], p.669
A Reaction
It seems that the axiom is essentially empirical, and it certainly makes an existential claim which seems to me (intuitively) to have nothing to do with logic at all.
Related Idea
Idea 15931 The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine]