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Jul
26
comment Is there a rule for uniform substitution of predicate symbols in FOL?
Substitution of predicates as you ask about is guaranteed to work by the fact that the logical axioms (I'm going to assume an axiomatic presentation) are defined to be substitution instances of certain schemata (and it is completely unrelated, as you guess, to quantification over predicates). I think what you're wondering is whether there's a specific rule of inference that lets you use that sort of substitution as a line of a proof, and there's generally not, for the reason that you can always write down a logical truth (wrt whatever logical system you're using) as a line in your proof.
Jul
26
comment Set-builder Notation
I don't know what the purpose of 1 and 2 are, but generally "$y\in \{x:\phi\}$" is just notation for "$\exists z(y\in z\wedge \forall x(x\in z \leftrightarrow \phi))$", so 3 is an immediate consequence.
Jul
25
comment What is the Viewpoint of Modern Logic?
@Doug: Why do you pick those two people? What is it about their work you're looking to contextualize?
Jul
25
comment What is the Viewpoint of Modern Logic?
What do you mean by "viewpoint"? Are there any specific features of logic such that you wonder what the "modern viewpoint" has to say on them? Are you interested in schools of philosophy of logic? What open problems are, according to general consensus, the interesting ones? General opinion on which tools for studying logic are best? I think the only fruitful thing to do when you have a question this broad is to read a lot of material on logic (possibly in several fields) until you can formulate some clearer questions.
Jul
25
comment What is the Viewpoint of Modern Logic?
I have to say I am not really clear on what the question is. Can you clarify more?
Jul
18
comment Order in writing composed morphisms
For what it's worth, Paul Taylor uses $f;g$ as a synonym for $g\circ f$ in Practical Foundations of Mathematics. I thought it was a nice convention to allow the more understandable ordering without it getting confused with the traditional ordering.
Jul
17
comment Cardinality of universal set?
No, NF itself is not known consistent, though Randall Holmes has a promising looking proof of Con(NF). Infinity is a theorem of NF, proved by Specker quite some time ago.
Jul
15
answered Cardinality of universal set?
Jul
11
comment Can all axioms of mathematical theories be expressed with predicate logic?
I think that's part of the issue: we don't really "informally axiomatize" theories because we want to be able to study said theories rigorously. It is also the case that there are structures that have no first order axiomatization (well orders, topological spaces), but which are nonetheless capable of being implemented in a theory that is first order (namely set theory), so for the purposes of your question I'm not sure how to count these. (I promise I'm not being difficult on purpose; I actually upvoted the question ;p)
Jul
11
comment Can all axioms of mathematical theories be expressed with predicate logic?
@Kyth'Py1k: My point was, what does "expressed in mathematics" mean; and if they were not expressed in a predicate logic to start with, what kind of "equivalence" with a sentence of predicate logic do you have in mind? I think the question is ill posed.
Jul
10
comment Can I soundly define a function which maps to itself?
@Asaf: That's one of those things perpetually on my "read up on it" list. Maybe this is the week...
Jul
10
comment Can I soundly define a function which maps to itself?
@AsafKaragila: Well, it obviously disproves foundation with $V\in V$, or that the set of all ordinals is a member of a member of the set of all ordinals; but it doesn't prove the existence of Quine atoms and such which have more the flavor of $f=\{(0,f)\}$. I'm sure you can squeeze such a thing out of a suitable permutation model, in any case.
Jul
10
comment Can all axioms of mathematical theories be expressed with predicate logic?
Axioms are sentences of some particular language, so I'm not sure what kind of notion of expressability you have in mind.
Jul
10
comment Can I soundly define a function which maps to itself?
@AsafKaragila: Actually, that's interesting... I can't think of a way off the top of my head to prove the existence of one of those in NF. Probably I'm being daft, it certainly seems like it should be lurking in there somewhere...
Jul
10
comment Can I soundly define a function which maps to itself?
Assuming $\mathsf{NF(U)}$ as your set theory, you can find a function that takes itself as both an input and a value in a very silly way: the identity function on the universe exists.
Jul
10
comment Set Theory (Example of Set)
The issue you're worrying about is whether the glasses are individuated by their physical properties. But you needn't have any non-trivial criterion for individuation for them to be, in fact, distinct objects. Incidentally, I loathe the "well-defined, distinct" qualifier I see so often; it seems like it's there to clarify, but it doesn't actually tell you anything useful...
Jul
8
comment Can all theorems be deduced directly from the ZFC axioms?
I'm not sure what the question is here. A theorem is only a theorem of some theory. ZFC trivially proves exactly its own theorems.
Jun
29
comment How do I find the type of relation on an infinite set?
(Actually I misstated things in my above comment; the second term of the second pair needn't be $\{\{a\}\}$, but it will still turn out that $a$ will need to be a member of every $x\in\mathcal{P}(A)$ such that $\{a\}\in x$, which leads to the same problem.)
Jun
29
comment How do I find the type of relation on an infinite set?
For transitivity, note that the question only arises when you have pairs of the form $(a,\{a\})$ and $(\{a\},\{\{a\}\})$. For the relation to be transitive it needs to be the case that $a\in\{\{a\}\}$ for all $a$. (The general methodological note here is not to worry about the picture of a list of stuff in curly braces, but to go back to the logical conditions that defined your sets.)
Jun
26
comment Do comonads always induce cosimplicial objects? Vice-Versa?
What one specifically gets from a comonad $L$ on $\mathcal{C}$ is a functor $F:\mathcal{C}\to \mathcal{C}^{\boldsymbol{\Delta}^{op}}$. So I doubt one could generally take a single simplicial object and get a comonad back out of it since, as you suppose, that's a lot of information to ask from one simplicial object. I have no idea about obtaining a comonad from, say, some sufficiently nice subcategory of $\mathcal{C}^{\boldsymbol{\Delta}^{op}}$, but I hope someone has an answer to that!