ZFC inconsistent due to Burali-Forti paradox


The Burali-Forti paradox makes ZFC inconsistent thus proving colin leslie
deans claim that mathematics ends in meaninglessness


http://en.wikipedia.org/wiki/Burali-Forti_paradox

suppose that we associate with each well-ordering an object called its
"order type" in an unspecified way (the order types are the ordinal
numbers). The "order types" (ordinal numbers) themselves are well-ordered
in a natural way, and this well-ordering must have an order type Ω. It is
easily shown in naïve set theory (and remains true in ZFC but not in New
Foundations) that the order type of all ordinal numbers less than a fixed
α is α itself. So the order type of all ordinal numbers less than Ω is
Ω itself. But this means that Ω, being the order type of a proper
initial segment of the ordinals, is strictly less than the order type of
all the ordinals, but the latter is Ω itself by definition. This is
absurd!ie a contradiction

thus demonstrating deans claim that mathematics ends in meaninglessness





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