Editing 2610: Assigning Numbers
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− | Notice that the truth of Gödel's statement does not depend on any particular set of axioms, and adding axioms (such as "Gödel's particular statement is true") only opens up new iterations of the statement which cannot be proven based on the expanded set of axioms (A statement such as "All statements of a similar nature to Gödel's particular statement" is not precise enough to serve as an axiom.). As such, with a little more legwork, it can be proven that any logical system robust enough to accommodate arithmetic must necessarily contain facts that are true within the system but cannot be proven or disproven within the system. The importance of this result cannot be understated, as it upended the entire philosophy of mathematics. {{w|David Hilbert}}'s famous proclamation "We must know, we will know" is simply incorrect. | + | Notice that the truth of Gödel's statement does not depend on any particular set of axioms, and adding axioms (such as "Gödel's particular statement is true") only opens up new iterations of the statement which cannot be proven based on the expanded set of axioms (A statement such as "All statements of a similar nature to Gödel's particular statement" is not precise enough to serve as an axiom.). As such, with a little more legwork, it can be proven that any logical system robust enough to accommodate arithmetic must necessarily contain facts that are true within the system but cannot be proven or disproven within the system. The importance of this result cannot be understated, as it upended the entire philosophy of mathematics. {{w|David Hilbert}}'s famous proclamation "We must know, we will know" is simply incorrect. Unless of course Gödel's theorems are flawed in some way that 90 years of highly motivated mathematicians have been unable to find. |
The title text suggests that Gödel should perform such an analysis on different branches of mathematics, by calculating the average of all the fields' theorems' Gödel numbers. This is nonsensical for a number of reasons: | The title text suggests that Gödel should perform such an analysis on different branches of mathematics, by calculating the average of all the fields' theorems' Gödel numbers. This is nonsensical for a number of reasons: |