Carl Mummert

Marshall University

science.marshall.edu/mummertc

Age: 36

I work in mathematical logic. My main areas of interest are arithmetic, reverse mathematics, computability, and proof theory.
4h
comment Bound Variable and Free Variable, A Questions and one Example?
The correct terminology is "bound variable". There is such a thing as a "bounded quantifier", which is different.
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awarded Yearling
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awarded Yearling
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answered What are the L-sentences that are true in an empty structure?
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comment If $4=5$, then $6=8\,$ (yes or no?)
I think this would be better as a comment, as it is very short and does not really answer the question.
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answered If $4=5$, then $6=8\,$ (yes or no?)
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comment If $4=5$, then $6=8\,$ (yes or no?)
The subtlety here is the specific choice of numbers. For example, we know that $6 =8$ does not imply $4 =5$ in the ring $Z_2$, where $4 = 1 + 1 + 1 + 1$, etc. It just so happens that $ 5 = 4+1$, which lets us show that in any ring with $4 = 5$ we also have $6 = 8$. Things would be different if we assumed $4 = 11$ or $4 = 13$, since those are true in more rings.
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revised How fast was the Turing's machine for breaking the enigma code?
edited tags
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revised Show the following languages are not recursive
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revised Is there an infinite field such that every non-zero element has finite multiplicative order?
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comment Is there an infinite field such that every non-zero element has finite multiplicative order?
I haven't heard the term 'reunion' before, does it have any mathematical synonyms?
Apr
16
comment Reference request for the independence of $ \text{Con}(\mathsf{ZFC}) $.
As Asaf's answer alludes, consistency alone is not enough, because ZFC + $\lnot \text{Con}(\text{ZFC})$ is consistent and proves its own inconsistency.
Apr
16
comment Dependence of axioms of Zermelo set theory
I wonder whether any of this is in the old Foundations of Set Theory book by Fraenkel, Bar-Hillel, and Levy. Independence of axioms sounds like the type of thing they may have written about.
Apr
16
comment Dependence of axioms of Zermelo set theory
@Asaf: I think the idea of the question was that many of the axioms are equivalent in the presence of replacement, so we might be able to get more independence by looking at models that don't satisfy replacement. You are pointing out that the ones that are already independent over ZFC are also independent over Z.
Apr
16
comment Reference request for the independence of $ \text{Con}(\mathsf{ZFC}) $.
There is nothing special about ZFC in this regard. The assumptions needed will be the same as for any other theory, and the authoritative reference would just be any reference on the incompleteness theorems. If you are looking for a good book on those, I recommend the one by Peter Smith.
Apr
15
revised How show $ S \models \forall x ( \alpha \Leftrightarrow \beta)$?
edited tags
Apr
15
comment Is $\exp(x)$ the same as $e^x$?
@Mario Carneiro: $\exp(z)$ defined from a power series has no branches, of course - it's an analytic function. If you look through various complex analysis texts, you will find it is quite common to define $a^b$ in terms of an arbitrary branch of the logarithm (of course, the principal value is usually also defined). Few books make any special provision for when $a$ is positive real; they just define $a^b$ for all $b$ and nonzero $a$ in the same way.
Apr
13
awarded Nice Answer
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13
awarded Autobiographer
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13
awarded Supporter
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