[I'll update this post as I find things that are interesting in logic]
I'm actually familiar with some of the things here: https://www.youtube.com/watch?v=tpeeTHNVnxQ and am watching the second part now: https://www.youtube.com/watch?v=VOhGLnxJt4E. (Joel David Thomkins 'The Continuum Hypothesis and Other Set-Theoretic Ideas for Non-Set-Theorists), which makes me glad that I have taken courses in analysis.
I've glanced at some literature on the algebra of cardinals at the University of Iowa library which I would like to understand better.
I tried reading Hermann Weyl's The Continuum Hypothesis last summer and a lot of it went over my head. Sitting in on a formal logic course on Irving Copi's Symbolic Logic (I got the 5th Edition for like $9 online) has been helpful and I'm hoping to get back to Weyl's book eventually.
I've taken two analysis courses and two abstract algebra courses and have done homework assignments on the Riemann Zeta function, Gamma function, and have heard about elliptic curves in a lecture from a course that I was taking for a grade. This is farther than I ever thought I would get in math, so I am happy about that. What I like about formal logic is that it can give me a way back into math, with stronger foundations: Irving Copi's formal logic goes into calculus and algebra by the end, so I would have a way to "stay alive" in mathematics. If I am not accepted to any graduate courses in logic then I would be able to say that I have given it a good run in math and would feel accomplished in my work.
I did touch on some Set Theoretical basics with my '14-'15 Algebra I students, so I'm glad that they'll have some access to Set Theory.
I learned that DeMorgan's Laws for Logic give an analog for the way that I need to remember to explain semantic incompleteness. I had previously noted:
Week of 9/12/16: It is the case that in any language, there is always something that cannot be expressed. It is not the case that there is some thing that cannot be expressed in any language; for, to name it would be to invalidate the reason we were searching for it. - (On the notion of semantic completeness)
This can be expressed in formal logic as:
¬∃x∀yφ(x,y) = ∀x∃y¬φ(x,y)
I am thinking of the left side as: and the right side as:
"It is not the case that there is a subset of the "For every subset of the real line there
real line which is such that for every language, = exists a language such that it is not the
the specific subset of the real line is inexpressible" case that the subset of the real line is
inexpressible."
Now, the left side is what we were given from the notion of semantic completeness. It is interesting that this is logically equivalent to saying that for every subset of the real line, there is some language in which the subset can be expressed.
The error I was tending to make in natural language was to say the negative case of the Week of 9/12/16 Quote: "It is not the case that there is some thing that cannot be expressed in any language..." Using formal logic keeps this straight in my head.
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