finitarily

What are the finitarily acceptable methods of construction and reasoning?

Hilbert's Program Zach, Richard 2003

Hilbert and Bernays (1934) give the only general account of finitary contentual number theory; according to it, operations defined by primitive recursion and proofs using induction are finitarily acceptable.

Hilbert's Program Zach, Richard 2003

By this he means that the contradictory proposition that there is a numeral n for which 1+n ‰  n+1 is not finitarily meaningful.

Hilbert's Program Zach, Richard 2003

An ordinal analysis of a system T is given if: (a) one can produce an ordinal notation system which mimics the ordinals less than some ordinal αT so that (b) it can be proved finitarily that the formalization TI (αT) of the principle of induction up to αT implies the consistency of

Hilbert's Program Zach, Richard 2003

P, P itself (or the finitary proposition it expresses) must be finitarily provable.

Hilbert's Program Zach, Richard 2003

To count as a finitary consistency proof, the operation itself must be acceptable from the finitist standpoint, and the proofs required must use only finitarily acceptable principles.

Hilbert's Program Zach, Richard 2003

This statement is finitarily acceptable since it “serves merely to abbreviate the proposition” that either p + 1 or p +

Hilbert's Program Zach, Richard 2003

This is finitarily acceptable as long as the characteristic function of such a property is itself finitary: For instance, the operation which transforms a numeral to 1 if it is prime and 11 otherwise can be defined by primitive recursion and is hence finitary.

Hilbert's Program Zach, Richard 2003

Detlefsen presents several lines of defense, one of which is similar to the one just described: arguing that a version of the ω-rule is finitarily acceptable, although not capable of formalization (however, see

Hilbert's Program Zach, Richard 2003