starkid wrote:
Furthermore, my problems with mathematical logic don't involve writing proofs. There is no reason for me to write mathematical logic proofs. I'm reading the books and working on understanding the proofs in them because I want to understand the logical foundations of math, not, for example, topology or abstract math taken by themselves. The mathematical logic proofs differ from the proofs in these more mainstream sub-disciplines in that they are more fundamental and are more directly based on pure symbolic logic. They are not and cannot be proved with the reasoning used in mainstream math proofs because their whole purpose is to develop that reasoning. For example, one of the theorems in my textbook is:
For every u, for every v, P implies that for every v, for every u, P.
In other words, it is proved that changing the order of universal quantifiers (u and v) has no effect on the meaning of the doubly quantified statement (P). This fact is taken for granted in mainstream math, but, with mathematical logic, it cannot be used to prove other things until it has itself been proven, simple though it is.
This one seems to be trivially obvious to me. Essentially, if x∈U and x∈V, then x∈U∩V=V∩ U.
The order should not matter as long as the number of quantifiers is countable.
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The comment about "cannot be used to prove other things" is a rather interesting comment. I have never had much of an understanding of when something was done mathematically and of putting developments in their proper order.
In one class many years ago someone asked the prof about a point that involved the order in which something was done. In particular, he pointed out that such and such theorem was not proven until the middle of the 20th century (in the 1950s, I think) and asked how a great deal of math that has been around for much longer could have been done without that proof. The prof's answer was that prior to that proof, we kind of danced around that missing proof.
27 May 2015, 8:39 pm
