Timeline for Meta$^{n{-}th}$ mathematics
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 14, 2014 at 7:59 | comment | added | Jesse Elliott | I'm not sure I agree. Con(ZFC) is a (probably true) statement to the effect that a certain complicated Diophantine equation has no integer solutions, expressed as a statement of PA. `ZFC is consistent' is the metamathematical statement that ZFC is consistent. They happen to be equivalent. Con(ZFC) is actually many statements, since it depends on what Godel numbering assignment you choose. So if they are all the same statement, then you are implying that all of the Diophantine equations you can get by different Godel number assignments are all identical, which is definitely not the case. | |
| Oct 13, 2014 at 11:45 | comment | added | Joel David Hamkins | @JesseElliott I believe that the distinction is not as clear as your remark suggests. For example, I believe that many people would say that both of those statements are meta-mathematical, and others would say that those are the same statement, with the same meaning, even though they are said differently. | |
| Oct 13, 2014 at 9:47 | comment | added | Jesse Elliott | It is clear that there are mathematical statements that are not metamathematical and vice versa. For example, the statement `ZFC is consistent' is purely metamathematical, while its translation Con(ZFC) via Godel numbering is a mathematical (in fact, arithmetical) statement. Is not the distinction a valid and useful one? | |
| Jun 28, 2014 at 12:46 | comment | added | Joseph O'Rourke | I have learned a great deal from this and the other responses and comments---Thanks! | |
| Jun 28, 2014 at 12:30 | vote | accept | Joseph O'Rourke | ||
| Jun 28, 2014 at 3:39 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
added 88 characters in body
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| Jun 28, 2014 at 3:26 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |