1. lord help me 2. guys i need help with part b of this question...

Demonstrate with a truth table that the chain rule in logic is valid: p → q , q → r ⇒ p → r .
b Using a as the basis, prove by induction that if n ≥ 3,
p1 → p2, p2 → p3, ..., pn-1 → pn ⇒ p1 → pn

so i know that i have proved by truth table that p1->p3 here as my base case, but now i dont know how to prove by induction the n-1 and n case, please guide me????
Roughly, the proof is almost immediate- I make it seem a little verbose below but the proof itself is just a few basic steps. As with a lot of mathematical induction proofs, some wonder "huh?, that's it?". This is because of the power of the tool of mathematical induction, and people tend to be skeptical because you assume S[k] is true at one point in an induction proof....but this step is totally legit and confirms the chain is indeed unbroken. When you see mathematical induction, think dominoes. You have a long line of dominoes (imagine it being infinitely long, but with a beginning). You can deduce that the 5,342nd domino is knocked down if you know only the following two bits of information: if any domino is knocked down, the next one in the line is knocked down and the first domino is knocked down (the base case). There is nothing special about the 5,342nd domino here: any domino in the line will be knocked down because of the 2 bits of info above, even the 7,343,232,564,346,246,349,222,985,384th domino. In the same way we can deduce a statement S[n] is true for all natural numbers n, if we simply know for every natural number k, S[k] is true implies S[k+1] is true, and  S is true.

Anyway, to the problem.
With mathematical induction, in addition to the base case, you must show that for every integer k >=3, that if S[k] is valid, then S[k+1] is valid, where S is the statement above in b.

If you assume S[k] is valid for arbitrary integer k>=3, then you essentially have the above statement in (b) holding for that particular k [just put k where n is].

You need to show S[k+1]: p1->p2,...,pk-1->pk,pk->pk+1 ⇒ p1->pk+1 holds. The only way this doesn't hold if the left side can be true, and the right side false. So we assume the left side is true, and see what happens.

If the left side is assumed true, all the single implications on the left side must be true. In particular, the first k-1 implications p1->p2,....,pk-1->pk are true, so we have p1->pk being true because of our assumption S[k] above.

So we know p1->pk,pk->pk+1 are both true [the latter because remember we are assuming the entire left side of S[k+1] is true. But from the basic general case shown in [a], it follows p1->pk+1 is true [we know that p1->pk,pk->pk+1⇒p1->pk+1 is valid from part a]. So there is no way for the right side of S[k+1] to be false with the left side true. Thus S[k+1] holds and we are done.

Note there are some subtleties here with material implication vs logical implication, but need not go into that. 3. Another way to look at this, although this is not how you were asked to prove it:

A ⇒ B is not valid if B can be false and A true.

Your B is p1->pn. This is false exactly when p1 is true and pn is false.

Your A is p1->p2,....,pn-1->pn. This is true exactly when every implication here is true.

Is there a way p1 can be true, pn be false, and every implication in A be true?

Assume p1 is true and pn is false. Let's see what would have to happen for A to be true.
-since p1 is true, p2 must be true (otherwise the first implication would be false, and A would be false)
-since p2 is true, then p3 must be true, (otherwise the second implication would be false, and A would be false)
-etc. since pn-2 is true, then pn-1 must be true.
In short, all pk, k=1,2,...n-1 must be true.
-But this would mean pn-1 is true and pn is false (our original assumption). The last implication would be false, making A as a whole false.

There is no way to make A be true, with B being false. Thus the logical implication A ⇒ B holds. 4. Wondering if any of u phaggots were aware of the new identity discovered for linear algebra recently that relates the eigenvalues of a hermitian matrix to its eigenvectors. A very surprising result that should have been discovered a hundred years ago, but somehow everyone missed it! Interestingly, it was a group of physicists who discovered the identity while working on neutrino scattering calculations.

htt ps://arxiv .org/abs/1907.02534

htt ps://terrytao.wordpress.c om/2019/08/13/eigenvectors-from-eigenvalues/ 5. Does this polynomial resemble anything to anyone? I derived an expression for the radiating magnetic dipole/electric quadrupole in the intermediate zone (long wavelength limit) and came across this expression. It has an uncanny resemblance to spherical hankel functions (of the first or second kind) but it's not quite there.  Maybe it can be cleverly constructed from them through linear combinations? (I'm not seeing it though) As always help is appreciated in advanced. 6. It seems like the first function above can be constructed merely by multiplying h_2^(1) (z) by (-i)..........or maybe I'm crazy.

The last expression I posted might be a little harder to write in terms of spherical hankel functions but I have an idea.

Edit: NVM I'm wrong. 7. who is the best mathematician of all time? 8. I'm staying with Geometry more or so myself nowadays due to the fact that I really enjoy creating / building stuff.

Plus it's pretty simple / easy for the math realm. So can peacefully enjoy it without getting stressed usually. 9. who is the best mathematician of all time?
I guess the standard answers here will be Newton, Euler, Gauss, Archimedes.

It's a question of opinion, not entirely different than who is the best artist of all time, or musician, or novelist.

As a fan of set theory and mathematical logic, I have always been amazed at the original creativity of Cantor and his set theory, and Godel with his fundamental contributions to logic, though these names won't find themselves on any typical list of greatest mathematicians of all time. 10. I guess the standard answers here will be Newton, Euler, Gauss, Archimedes.

It's a question of opinion, not entirely different than who is the best artist of all time, or musician, or novelist.

As a fan of set theory and mathematical logic, I have always been amazed at the original creativity of Cantor and his set theory, and Godel with his fundamental contributions to logic, though these names won't find themselves on any typical list of greatest mathematicians of all time.
i never understood the incompleteness theorem, and the bro-tier understanding i have is that:

logic based on a prior axioms can never be complete?

Gödel defines a book-keeping device, a well-ordering of all tuples of variables arising from a need to satisfy φ as dictated by (Q). For example, if (Q)φ is ∀x0∃x1ψ(x0, x1), we list the quantifier-free formulas ψ(xn, xn+1). (Or more precisely, finite conjunctions of these in increasing length. See below.) Then in any domain consisting of the values of the different xn, in which each ψ(xn, xn+1) is true, the sentence (Q)φ is clearly true. A crucial lemma claims the provability, for each k, of the formula (Q)φ → (Qk)φk, where the quantifier free formula φk asserts the truth of ψ for all tuples up to the kth tuple of variables arising from (Q), and (Qk)φk is the existential closure of φk. (See the example below where the definition of the φk′s is given.) This lemma is the main step missing from the various earlier attempts at the proof due to Löwenheim and Skolem, and, in the context of the completeness theorem for first order logic, renders the connection between syntax and semantics completely explicit.

Let us consider an example of how a particular formula would be found to be either satisfiable or its negation provable, following Gödel's method: Consider φ = ∀x0∃x1ψ(x0, x1), where ψ(x0, x1) is quantifier-free. We show that this is either refutable or satisfiable. We make the following definitions:

φ0 is the expression ψ(x0, x1)
φ1 is the expression ψ(x0, x1) ∧ ψ(x1, x2)

φn is the expression ψ(x0, x1) ∧ …∧ ψ(xn, xn+1).
dude i cannot understand this language, i only took diffy q's and never any modal logic or whatever this is. is there a modal logic book for morans i can look at? 11. i never understood the incompleteness theorem, and the bro-tier understanding i have is that:

logic based on a prior axioms can never be complete?

dude i cannot understand this language, i only took diffy q's and never any modal logic or whatever this is. is there a modal logic book for morans i can look at?
Godel's theorems (there are 2) demonstrate certain limitations of formal axiomatic systems having particular properties. An important thing with this is: the theorems dont apply to just any old system in mathematics, but rather formal axiomatic systems that are capable of modelling ordinary arithmetic. Typical examples: ZFC (axiomatization of set theory), first-order Peano arithmetic, etc.

The first Godel theorem says that if you have an effectively axiomatized formal system that can model arithmetic, then that system is either inconsistent or incomplete (it cannot be both consistent and complete). This has far reaching ramifications: essentially it is saying that in a consistent system, there will always be statements that are not provable within the system, whose truth is undecidable: you cannot create a set of axioms that lead to all mathematical truths, in a sense (as earlier mathematicians like Hilbert, Russell, and Whitehead, were attempting to do, until Godel said "nope").

Godel did this by constructing a "Godel sentence", which takes advantage of the self-referential paradox... the incompleteness theorem at its heart really is hinged on this idea of the self-referential, like the old "this statement is false", but in Godel's case, it takes the form "this statement cannot be proved".

His second incompleteness theorem goes on to say that you cannot prove consistency of a system X from within X.

Some concrete applications: for the 1st incompleteness theorem, if we take ZFC to be consistent, then we know that there must be statements which neither themselves nor their negations are provable within ZFC. An example of such is the continuum hypothesis, that there is no infinite set with cardinality strictly between that of the integers and real numbers- which has been proven independent of ZFC. We could merely add the CH as an additional axiom, but then we are still left with the same problem in the new formal system: if consistent, there will still be another statement that is undecidable from within the system. Godel's theorem says you will never get away from this fact.

An example of the second theorem: you cannot prove ZFC itself is consistent.....from within ZFC. Most mathematicians expect it is consistent, and indeed no examples of inconsistency have been demonstrated to date (you could prove inconsistency by merely exhibiting a contradiction).

The above is just a brief intro, Godel arrives at his incompleteness theorems ingeniously by using a sort of "code" to represent sentences in his systems by numbers. To learn about his theorems, you can go straight to the source...

https://store.doverpublications.com/0486669807.html

(highly, highly, not recommended. This is a dense, technical work)

The best exposition of the subject i've seen is actually in the form of a puzzle book by the late great Raymond Smullyan (the masterful "wizard" of mathematics). He wrote many puzzle books of logic, but they arent just any logic puzzles, but rather ones with deep significance in terms of mathematical logic. He arrives at Godel's incompleteness theorem by the end of the book below, it is really an awesome book.

https://store.doverpublications.com/048647027x.html 12. dope thanks. i cant obv get my head around it bc the words are practically foreign to me. but i will check out the book.

i can image [1,2,3,....] containing [1.0,1.1,1.2...] is consistent but incomplete? based on a CH axiom.

thanks for taking the time 13. dope thanks. i cant obv get my head around it bc the words are practically foreign to me. but i will check out the book.

i can image [1,2,3,....] containing [1.0,1.1,1.2...] is consistent but incomplete? based on a CH axiom.

thanks for taking the time
Yeah, you make a good point about the jargon. Some basic terms must be established before things like the incompleteness theorems can even be talked about: terms like consistent, complete, independent, etc.

These terms are not applied to sets (I may be misinterpreting your post above), but rather to, for example, axiomatic systems.

Consistent: A system which contradictory statements are not provable in the system. Say you have some axiomatic framework X underlying whatever mathematics you are doing. You eventually prove 1) that pi is rational and 2) that pi is irrational within this system X. Then X is not consistent, you have a bona-fide contradiction. Inconsistent systems are essentially worthless.

Independent: A statement is said to be independent of a list of axioms if neither it nor its negation can be proved from those axioms, and an axiomatic system in which all axioms are independent is called independent. A famous example is Euclid's 5th postulate, the parallel postulate*, which many hopelessly tried to prove for almost 2000 years, until it was realized it was independent of the other postulates (and this had important ramifications for non-Euclidean geometry). Euclid was right to just add it as a postulate in his system.

*: the parallel postulate states that two lines, on the side of an intersecting line which makes angles less than 2 right angles, eventually meet (on that side of the line). This is a rather complicated statement to take as a postulate, so it bothered many people...but alas, it is rightfully a postulate. The more common rendition is playfair's axiom: if you have a line, and a point not on that line, then there is at most one line going through that point that is parallel to the original line (and hence, exactly one: Euclid Book 1 Proposition 31, which relies only on the other postulates and not the 5th, demonstrates that a parallel line does exist).

Complete: A system in which all statements are provably true or false can be called complete. Consider Euclidean geometry, but with just the first 4 postulates and not the parallel postulate (called "absolute geometry"). This cant be said to be complete, as the truth of the parallel postulate itself is undecidable. 14. that im unfamiliar with terminology is a 'me' problem. im really interested in takin these more complex ideas and puttin them into lay terms (for myself), but im just reading through this page on hilberts third problem, and i don't see how these things can be simplified. critical ideas and points are attenuated in the dumbing down. i am reminded of feynman's magnet video. 