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Yeah I think the idea is that unrestricted comprehension could be used in place of the more basic axioms, if consistency be damned. Of course, the inconsistency of unrestricted comprehension means that we can't really reason with it seriously but we can play along for intuition perhaps. Not sure, but I think unrestricted comprehension: - Doesn't grant Extensionality - Doesn't grant Foundation - Grants Specification - Grants Pairing - Grants Union - Grants Power Set - Grants Replacement - Grants Infinity - Doesn't grant Choice We can use different formulas phi along with unrestricted comprehension to quickly get the existence of basic things. Pairing of a and b: phi_Pa(x,a,b): Forall y (y in x)<->(y=a Or y=b) {a,b} = {x | phi_Pa(x,a,b)} Union of a: phi_U(x,a): Exists y (x in y)&(y in a) U(a) = {x | phi_U(x,a)} Powerset of a: phi_P(x,a): Forall y (y in x)->(y in a). P(a) = {x | phi_P(x,a)} Replacement for psi(x,y,A,z_1,...,z_n): phi_R(y,A,z_1,...,z_n): Exists x [(x in A) & psi] R(A,z_1,...,z_n) = {y | phi_R(y,A,z_1,...,z_n)} The case of interest is psi a function from x to y. It grants infinity too, but now I'll have to prove by contradictions, which brings the silliness of all this to light. But I need to assume that there exists a set b. The pairing of b with itself is {b}, so singletons exist. If Foundation is true, then b and {b} are different. Let U be the set of all sets: {x | x=x}. P(U) contains {x} for all sets x in U. These cannot all be elements of U by induction in combination with the fact that no x is {x}. Contradiction. Unrestricted comprehension may not make much sense as is. But it's similar to what you can do in something like a Grothendieck universe, say, V_k for an inaccessible cardinal k. The (consistent) axiom of specification when applied to V_k as a set then appears like unrestricted comprehension on the universe V_k but where the returned collection is generally a subset of V_k, not necessarily in V_k. Similarly if we're in a class theory like NBG. Universal comprehension then just has to be modified to allow the returning of proper classes. In fact, this is basically how we use classes even in ZFC: they are collections defined by unrestricted comprehension!

So cool :)

Brown Lisa Brown Margaret Wilson Lisa

Amazing.

yeah but you can't keep going--you only think you can . Give me one example of infinity . A concrete example .

{2,3,5,8} -> 2^2*3^3*5^5*7^8. (woops I wrote the wrong example for the godel coding 9:32)

Hello

Great visuals!

th-cam.com/video/WszGQrPMnd0/w-d-xo.html

Sounds good Erin, Aaron.

I am a borderline "Cantor Crank," as I have an intuitive dislike for some related things. Borderline, because I believe that, if there was some problem with Cantor, some brilliant and genuine mathematician would have demonstrated the flaw. Thus, I accept that the conclusions of the Cantor/set theory crowd flow rigorously from the axioms with which they work. Having said that, I think I understand that the conclusions I dislike arise from the Axiom of Infinity. The problem is that this is where we draw a boundary around infinity; we put it into a "box." Specifically, by calling it a "set," it seems we then can discuss the cardinality of that set. That cardinality is "infinity", and with the Axiom of Infinity, it seems the next thing you know you have an infinite hierarchy of bigger and bigger infinities. Probably the first objection of every Cantor crank is rejecting the statement that there are infinitely more irrational numbers that rational numbers. Even though, for two irrational numbers x and y, there exists a rational between the two (can be shown by truncating the binary representation of x and y at their point of difference, and taking the first rational. e.g. x = 1....11... and y= 1....10... and taking the rational as z=1....11. x>r>y qed.) This is only not possible if x and y have the same binary representation, but that would require them to be equal in value, which suggests they are not unique. So, I will accept that set theory somehow addresses this apparent objection. However, I really, really don't want to accept the conclusion (since I like working with numbers in an intuitive fashion.) I am pretty confident that the statement "more irrational than rational numbers" rests upon the Axiom of Infinity, so I feel that I can't accept that Axiom. Question: What do I lose if I reject this axiom? Does calculus still work?

how the hell did i get here? no idea i already knew binary numbers but stood until the very end, love the way you talked about it and the overall randomness of the video

Amazing night in JC. The operatics were so wonderful - how great they art. This guy said he quit his job in finance to pursue his singing passion and he does not regret it! Totally agree - DO quit your dayjob!

well, i got some good news =)

Thanks for uploading! If you have a second, I tried my hand at a cover of Jason Isbell. As a fellow music lover, I'd love if you'd check it out. It's on my page. Hope you like it!

Nice. Nice. Nice.

I heard "Piranha outside the happy family" and it made me smile. :-)

Jai guru deva!

Hey... just discovered you through your hand shake problem video..