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Full Idea
The notion of finitude is explicitly 'built in' to the systems of first-order languages in one way or another.
Gist of Idea
The notion of finitude is actually built into first-order languages
Source
Stewart Shapiro (Foundations without Foundationalism [1991], 9.1)
Book Ref
Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.238
A Reaction
Personally I am inclined to think that they are none the worse for that. No one had even thought of all these lovely infinities before 1870, and now we are supposed to change our logic (our actual logic!) to accommodate them. Cf quantum logic.