Law of excluded middle: mathematical philosophy

Is it a contradiction, if

• mathematical objects are for me "mental constructions" -- like in intuitionism

but

• I accept classical mathematics and the Law oft excluded middle

?

• If you accept the Law of the Excluded Middle, you have to come up with some other solution to Russel's paradox. Rejecting Brower's suspension of 'not' is not rejecting his idea that mathematical operations are psychological habits and not laws of nature. But it is not a good idea to hang onto a reasoning habit that leads you directly into paradox. – jobermark Jul 27 '15 at 16:51
• @jobermark What does LEM have to do with Russell's paradox? The derivation of a contradiction from naïve comprehension is intuitionistically valid. – J.P. Jul 27 '15 at 19:58
• @J.P. The paradox involves three steps. You have to say "1) it is unambiguous what is meant for x not to be in x (free self reference). 2) construct the set of all x with x not in x (unbounded universal quantification). 3) either x is in x or x is not in x (LEM) and you get a contradiction." So removing any of the steps solves the problem. Set theory cripples unbounded universal quantification, intuitionism undercuts LEM, and ramification or sequencing of types limit free self-reference. Each is a solution to this problem. – jobermark Jul 27 '15 at 21:00
• In intuitionism it is simply not true that either x has to be a member of itself or it has to not be a member of itself. It can be undeterminable whether x is a member of itself. "tertium datur". – jobermark Jul 27 '15 at 21:07
• @jobermark It's just false (and therefore misleading) to say that the paradox involves your step (3) (LEM). Let r be the Russell set. Suppose that r∈r, then ¬(r∈r), which is a contradiction, so by intuitionistically valid reductio, ¬(r∈r), and we discharge the assumption. Then, r∈r, and we have a contradiction, independent of any assumptions. At no stage is LEM used, all is used is the intuitionistically valid principle that if you can derive a contradiction from p, then you can infer ¬ p. See this question. – J.P. Jul 29 '15 at 7:11

It depends on what you mean by "mental construction":

• If you mean mental construction, i.e. just in mind, then there is no contradiction.

• If you mean mental construction, i.e. constructible step by step by a finite algorithm, then it is a contradiction.