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'Moo' is just another suggestive label for the 'true bucket'. You can label it number 1 if you like. Mathematicians aren't defining the 'bucket', they are only determining the conditions of when things belong in that 'bucket'.
I'm sorry but your simple sentence is just wrong. I suppose you're not aware that Kurt Godel proved that any formal system is capable of producing contradictions, if you use the exact same line of thinking you outlined in your simple sentence. Your definition of truth eventually leads to paradoxes.
I don't recall Goedel doing any such thing. His first theorem of incompleteness theorem showed that certain kinds of adequate and consistent formal systems must include truths (i.e., derivable truths) that were also unprovable (i.e., they are incomplete). His famous S sentence: "this sentence is not provable" is true but not provable. His second theorem of incompleteness basically said that for certain consistent formal systems the same formal system cannot be used to prove its own consistency.
There has to be something simpler and more effective. If not, how could Godel do what he did?
Mike, you've got to get over this idea that truth is defined in mathematics. It isn't. Mathematics, like all formal systems, simply labels the results of all decideable statements with labels like true/false (or in case of multi-value logic, there are more categories for decideable statements). Read Graham Priest's excellent article on-line regarding dialetheism (link: http://plato.stanford.edu/entries/dialetheism/ ). If truth were defined in mathematics, then it would not be possible to construct other truth-conditions besides true and false. That is, mathematics would directly contradict a dialetheistic approach, and why would logicians even bother?
As for how Goedel did what he did, he used some clever logical techniques (e.g., diagonal argument and Godel numbering). For example, in the first incompleteness theorem he showed that his famous sentence S can be constructed to show that if S is provable then it is false, but if it is not provable, then it is true. Since S can not be provable and false (i.e., false sentences cannot be proved to be true), therefore it means that there exists sentences in a certain kind of formal systems that are true but not provable.
Notice that this does not contradict my 'simple sentence' of mathematical truth. I mentioned that a true statement is one that is derivable under the rules of inference along with any defined and undefined terms, symbols, variables as allowed or included in the formal system in question. I didn't say a true statement is one that must be provable in the formal system. Had I made that mistake, then you would have gotten me. |
