It's also interesting that it is possible to adopt a 'countable only' philosophy that allows *countably* infinite sets, but not uncountable ones.
This system has the advantage of allowing construction of specific real numbers even though it does not allow the construction of the set of *all* real numbers. Also, it is possible to do a lot of the work with Turing machines and proof theory without the circumlocutions required in a purely finitist system.
The biggest problem with the 'countable only' system is that it does NOT allow the construction of the power set of a set: the set of all subsets of a set. All the rest of the standard axioms are satisfied, however, and this allows a great deal of math to be done.
In contrast, the finitist system uses all of the standard axioms except the axiom of infinity (which assumes the existence of an infinite set).
This system has the advantage of allowing construction of specific real numbers even though it does not allow the construction of the set of *all* real numbers. Also, it is possible to do a lot of the work with Turing machines and proof theory without the circumlocutions required in a purely finitist system.
The biggest problem with the 'countable only' system is that it does NOT allow the construction of the power set of a set: the set of all subsets of a set. All the rest of the standard axioms are satisfied, however, and this allows a great deal of math to be done.
In contrast, the finitist system uses all of the standard axioms except the axiom of infinity (which assumes the existence of an infinite set).
