Recent content by Simpl0S

Homework Statement Proposition. Every subset of a finite set is finite. 2. Relevant definitions Definition. Two sets ##X## and ##Y## have the same cardinality iff there is a bijection between ##X## and ##Y##. A set ##X## is finite iff there is a bijection between ##X## and ##\{1, ... , n\}##...

I think I expressed myself in the wrong way or you misunderstood me. I did not mean to draw a conclusion whether the view of finite or infinite is the correct one. I did not mean to say "finitist's math is nonsense". I meant his justification as that the natural numbers are finite is nonsense. I...

Wildberger says at around 11:28 that even if you were to build the most powerful computational machine you would not be able to compute a given large natural number and then draws the conclusion or raises the question that such a large number might not exist. I object there. This is not a proof...

Thank you for the reply. So if I want a deeper treatment of this subject then I have to look in the field of category theory? Any recommended books on that subject?

I am currently taking a course in discrete mathematics. The literature used is "Discrete Mathematics And Its Applications by Kenneth H. Rosen" 6th ed., or 7th ed. I have encountered most of the topics from that book. I.e. Logic, naive set theory, &c. What I have encountered also is the...

Put in those therms then I would say no, for the functions I have listed there is no such polynomial, thus making the trigonometric and exponential function (and their inverses), transcendentals. Thank you.

Thank you all for the replies and help. It clarified everything. I also skimmed forward to Spivak's chapter 19 "Integration in elementary terms" and there he presents what other books would label as "Integration techniques". So for now I have to deal with using the definition and the Fundamental...

I think I expressed myself wrong. Let me explain my problem. Spivak goes on and develops the intuition of the integral and then defines it as mentioned in post #1. After that he goes on and presents Theorem 2, which is a re-statement of the definition of integrals, because according to him, it...

@fresh_42 Thank you for the article. It was an enlightening read into my question and it was nice to see what comes after the Riemann integrals. Spivak also has an Appendix called "Riemann Sums" where he points out what a Riemann sum is. But even there Spivak does not use limits in his...

Hello. I finished working through Spivak's Calculus 3rd edition chapters 13 "Integrals", and 14 "The Fundamental Theorem of Calculus". By that I mean that I read the chapters, actively tried to prove every lemma, theorem and corollary before looking at Spivak's proofs, took notes into my...

In Spivak's Calculus, on page 121 there is this theorem Then he generalizes that theorem: I tried proving theorem 4 on my own, before looking at Spivak's proof. Thus I let c = 0 and then by theorem 1, my proof would be completed. Is this a correct proof? Spivak's proof for theorem 4...

Now that you put it in terms of distance it makes sense. But how does one develop the intuition to "see" what value for a variable one should choose when proving theorems? I figured the delta part out: Since the definition of a limit states, that "for all epsilon > 0, there is some delta >...

I downloaded the PF app. And I noticed that one can use it to upload pictures. The problem and part 1 of the proof: Part 2 of the proof:

Hello I am struggling with proving theorem 1, pages 98-99, in Spivak's Calculus book: "A function f cannot approach two different limits near a." I understand the fact that this theorem is correct. I can easily convince myself by drawing a function in a coordinate system and trying to find two...