wtf

I am seriously impressed! Note by the way that the number of digits is [math]\log_{10}(3^{5000}) = 5000 \log_{10} (3) = 2385.6062...[/math] according to Wolfram Alpha, which checks out.

That doesn't sound reasonable. I can think of two possibilities here: 1) You annoyed the teacher and (s)he is giving you something that will keep you busy a long time; or 2) You are supposed to find the remainder mod something. For example mod 4, every power of 3 is either 1 or 3 and you can easily figure out which ones are which. What do you think? Anything to my theory? Otherwise, get a box of pencils and a stack of paper. Or perhaps are you supposed to just estimate its order of magnitude? That would be another more reasonable request.

Can you show a fully worked example that takes some number and shows it to be probably prime or not? I don't need to see your code. Just a clear description of the algorithm. I would like to see a fully worked example that starts from some number and produces the result "probably prime" or not. I'd also like to see the statistical results of your research. Out of N numbers tested, how many were judged prime that were prime; judged prime that were not prime; judged not prime that were prime; and judged not prime that were not prime? N doesn't have to be large but perhaps up to 1000 or so would give us an idea if this is working or not. If it's working, then the fun is to figure out why! And I still don't know what you mean by choosing 25 and 7 at random. What number are you testing for primality and what does it mean to randomly choose a number? It's very difficult to randomly choose natural numbers since there's no uniform probability distribution on them. So just to make this concrete, can you run your algorithm, for say n = 15 and n = 11, and say whether your method shows them likely prime or likely not prime. ps -- Did I understand you to say you went a lot higher and got 361 out of 2745 correct? That's not a very auspicious start. Though of course no finite amount of experimentation can ever prove the truth of anything in number theory. But is that what you said?

"f(25, 7) = 3" Why did you start with 7? Why not start with 25 mod 11 for example? Also I'm not sure exactly which number you're probabalistically proving prime. 25 or 7? Can you make this a little more clear? I admit I didn't follow your method in detail, I got lost when you pulled 7 out of the air without explanation. If you apply your method to the first million numbers, how many "probable primes" are actually prime and how many are composite? In other words how many hits and how many false positives? Have you determined your method to be useful for large numbers where trial division isn't? Mod operations are more efficient than division so perhaps so, if it works. But I don't understand what exactly you're doing. Is 25 a probable prime?

Ok, [latex]T\ [/latex]is a constant. What kind of constant? Integer, real, complex? Quaternion maybe? I'm guessing it's a real but you should not make your readers guess. Better to say: Let [latex]T \in \mathbb R[/latex]. When you say that the domain is the integer multiples of [latex]T[/latex], then that's all we know about f. As far as we know it's not defined anywhere else. But I don't think that's what you mean. Rather, I think you mean that [latex]f[/latex] is defined on ALL the reals and happens to be zero on integer multiples of [latex]T[/latex]. Is that right? If so: Let [latex]f : \mathbb R \rightarrow \mathbb R[/latex] with [latex]f(nT) = 0\ \text{if}\ n \in \mathbb Z[/latex]. Now is [latex]f[/latex] continuous? Differentiable? What else are we assuming about [latex]f[/latex]? If you have any other unspoken assumptions, they should be stated clearly. You should write [latex]f(x)[/latex] on the left. Otherwise we don't know what the independent variable is. Is it [latex]x[/latex]? Is it [latex]T[/latex]? Or maybe it's [latex]\pi[/latex]! Since [latex]x[/latex] is the independent variable, we write [latex]f(x)[/latex]. I didn't look at the rest but hopefully you see what needs to be clarified. The main thing is that the domain is ALL the reals, and the zeros happen to be integer multiples of [latex]T[/latex], assuming that's what you mean. And as a general principle, try to eliminate any ambiguity in the mind of the reader. ps -- You have a subtle error in your logic. You have [latex]g(x) = x^2[/latex] and later you ask us to consider [latex]g^{-1}[/latex]. But [latex]g^{-1}[/latex] does not happen to be uniquely defined. So you have to either tell us what [latex]g^{-1}[/latex] is, or show that your argument is independent of the choice of a particular one-sided inverse of [latex]g[/latex]. Maybe I better say a few words about a common point of confusion. The word inverse has two different meanings when it's applied to functions. If a function [latex]f[/latex] is invertible, that means (by definition) that there exists some specific function that uniquely inverts it. (I'm omitting the technical details here). But [latex]x^2[/latex] is not invertible. Rather for each point in the range, there are either one or two points in the domain that get mapped to that point in the range; and we can create "a", not "the" inverse by making a choice from each of the inverse images of points in the range. But this will only be a one-sided or "partial" inverse. So you can't just use the notation [latex]g^{-1}[/latex] because it's not uniquely defined. Here's more than you ever want to know about it, but you should give this page a look: https://en.wikipedia.org/wiki/Inverse_function Pay particular attention to the section on partial inverses. https://en.wikipedia.org/wiki/Inverse_function#Partial_inverses

Notation is hard to follow. A few confusions on my part: * What is [latex]T[/latex]? * What can it mean for [latex]f : \mathbb Z T \rightarrow 0[/latex] to be periodic? It can't be anything other than the zero function. It's trivially periodic with every number in the domain being a period. * Since [latex]f[/latex] is the zero function, [latex]f \circ g^{-1}[/latex] must be zero anywhere it's defined. Can you clarify your intent here?

2 is an odd prime. A very odd prime.

I'd use Zorn's lemma.

There was a conference in December. Results were somewhat disappointing. Nobody understands the proof but some progress is being made. http://www.nature.com/news/biggest-mystery-in-mathematics-in-limbo-after-cryptic-meeting-1.19035 http://www.math.columbia.edu/~woit/wordpress/?p=8160 http://mathbabe.org/2015/12/15/notes-on-the-oxford-iut-workshop-by-brian-conrad/ https://www.maths.nottingham.ac.uk/personal/ibf/files/symcor.iut.html

That can't actually be right. Unicode currently maps over 109,000 characters. Surely you can see that if you took all the English, Chinese, Japanese, Arabic, Indian, and Russian alphabets together you'd have far more than 256 characters, and I'm leaving out lots of alphabets. Of course that doesn't take away from your invention of a 256 character alphabet. But there are plenty of natural language characters in the world. http://stackoverflow.com/questions/5924105/how-many-characters-can-be-mapped-with-unicode

That's a very weird comment relative to what I wrote, which is perfectly standard terminology and couldn't be more clear. Are you making a joke? I really don't get you. Is that your definition of a surjection? Care to revise your statement?

John Cuthber was talking about [math]x^7 - x^5[/math]. Note the minus sign. That function is not injective. That means that it takes the same value at more than one point, so that it does not have an inverse unless you restrict the domain. Your function [math]x^7 + x^5[/math] does have an inverse. It never takes the same value twice. However, we can't solve for the inverse in terms of elementary functions or algebraic operations. I don't have a proof for that fact, only a meta-proof; namely, that if there were some closed-form or elementary solution, Wolfram Alpha would know it. Of course that's not really a proof of anything.

It's just that if some horizontal line intersects the graph at more than one point, then the function can't be invertible; simply because there are two x-values that produce a single y.

Totally agreed. Some functions are invertible and some aren't. It's the horizontal line test from analytic geometry.

I didn't understand your point about there being three values etc. Both x^7 + x^5 and its inverse are bijective from the real to the reals. Besides, you swapped the + for a -, that confused me too. In any event, there's no elementary inverse. Proof by Wolfram. If an elementary solution exists, Wolfram would know it. https://www.wolframalpha.com/input/?i=inverse%5By+%3D+x%5E7+%2B+x%5E5%5D ps -- It's clear that x^7 - x^5 is not injective. https://www.wolframalpha.com/input/?i=x%5E7-+x%5E5

Forgive my pedantry here but there is no set of all sets and this is a semantic point of importance to me. The reason I insist on this is because in order to get anywhere at all, we have to agree on the meaning of "set". In this thread we are discussing sets in ZFC. In ZFC there is no set of all sets. It's perfectly ok to think about the collection of all sets, the aggregate of all sets, the proper class of all sets, and as Xerxes notes, the category of all sets. But in math the word "set" has a very specific meaning and according to this universally accepted meaning, there is no set of all sets. As far as mind-blowingness, that's what we love about set theory! Transfinite set theory is a triumph of the human imagination. It allows us to reason rationally about infinite sets. It's a brand new discovery, only 140 years old. That's pretty recent, considering that Aristotle thought about these same issues. So it takes some work to get one's mind around it. You're in good company there. That's true. But lots of simpler things in math "never reach a conclusive boundary." For example say we have the set of counting numbers 1, 2, 3, 4, ... Then we can make the set of rational numbers 1/2, 1/4, 1/8, ..., 1/2^n, ... and this also is a set that "never reaches a conclusive boundary." Lots of things don't have boundaries. So we should not be unduly impressed. For example, we could think about the class of all sets as "being the limit" of larger and larger sets of sets in the same way that 1/2^n has the limit 0 but no individual term is zero. [This is an idle thought on my part, I don't know if set theorists think that way or not.] First, here is Russell's paradox. It shows that 1) You cannot form a set out of a property alone. For example "is a set" is a property. Russell's paradox shows that we MAY NOT form a set by saying, "The collection of all things that satisfy some property." If we allow that, we get a contradiction. 2) It follows as a corollary of (1) that there is no set of all sets. I don't want to get bogged down in the details of this proof in this thread right now. https://en.wikipedia.org/wiki/Russell%27s_paradox The post I referred to earlier is this one: http://www.scienceforums.net/topic/92101-what-is-the-minimum-number-of-properties-posessed-by-members-of-a-set/?p=900416 That's a long post in the middle of a long thread but inside there is an explanation of why Russell's paradox implies there's no set of all sets. Read the Wiki article on Russell's paradox first. My own opinion is that set theory doesn't get interesting until you consider infinite sets. But then I'd be insulting the combinatorialists, who do amazingly clever things with finite sets! But as far as transfinite set theory being "practical," there's no known use for it at the moment. If tomorrow morning some physicist finds evidence of an infinite collection of things in the universe, then transfinite set theory will become important in the real world. Till then, this is purely an abstract intellectual pursuit. Although I would point out that there is a connection between transfinite set theory and computer science. Turing and Gödel were all over these connections in the 1930's. So one should not discount the theoretical importance of infinite sets even in the real world. I gave that a very brief glance, is it mostly philosophical? It think it's helpful if we realize that we have two different concerns: (1) learning about elementary set theory the way modern mathematicians understand it; and (2) arguing about philosophy. I'm attempting (1) and perhaps my friend studiot is coming from a perspective of (2). We are in the math forum after all, so why not stick to math and open a thread in the philosophy forum for philosophical concerns. My two cents on that. I happen to love math philosophy but I do not confuse it with doing math.

There are non-wellfounded set theories, in which sets can be members of themselves. I don't know much about them. Perhaps they would be of interest to you. https://en.wikipedia.org/wiki/Non-well-founded_set_theory So then why not just tell me where you're coming from? Can you say more about what you mean? What general theory is ZFC a restriction of? On the contrary, I explicitly outlined exactly how numbers are defined as sets within ZFC. Not how I learned geometry, and certainly not the modern view. So at best, your knowledge of what Euclid originally said would be of historical interest. It doesn't actually bear at all on the fact that in ZFC, both "set" and "element of" are undefined terms.

Since the subject is ZFC, a set is an undefined term. That may well be a problem for some philosophers. But it's not a problem in ZFC, since "set" is an undefined term in ZFC, in the exact same way that "point" and "line" are undefined terms in Euclidean geometry. To be frank I do not understand where you're coming from. Set theory has served mathematics well for a century. I don't understand why you seem to be arguing against standard set theory when -- you'll pardon my directness -- you clearly don't have enough knowledge of set theory or philosophy to have an informed opinion. That's not a slur or an attack. If you are an expert at tennis and I've never played before, my skill level will be immediately apparent to you the moment you see me play. I don't understand why you're pretending so much knowledge that you don't actually have. Like I say that is not a personal remark. It's something I'm very curious about. That's a perfect illustration of what I mean. Russell's shoe and sock example illustrates why we need the Axiom of Choice. If we have infinitely many pairs of shoes, we may say, "From each pair choose the left shoe." There is a first-order definable property that tells you exactly which choice to make. In the case of socks, there is no such property and therefore we need a whole new axiom to allow us to choose an element from each set. AC is a pure statement of existence and gives no clue as to the nature of the elements being selected. Do you know of some different context or meaning for this example? I'm curious as to why you brought it up, since (as far as I know) it does not bear on the question at hand.

The problem with this approach is that now we have to ask the question, "What is a number?" If we are proposing set theory as the foundation of math, we can't just say, "Well, a set is a thing that contains numbers." Because now we have no definition for numbers. If my professor says, "Consider the set of real numbers," I am now justified in responding: "But how do you know there are any real numbers at all? What are they?" And now we've reverted ourselves back to the state of math in 1870 or so. We've just thrown out 140 years of mathematical progress. The solution in ZFC is to define the real numbers as particular sets. The real numbers are equivalence classes of sets of rationals, the rationals are equivalence classes of pairs of integers, the integers are equivalence classes of pairs of natural numbers, and the naturals are built up out of the empty set and the successor operation. Everything can be built up in terms of sets. In any event, OP asked specifically about ZFC. In ZFC there are only sets. Everything is a set. A set contains other sets and a set must be contained in some other set. In particular, numbers are sets. Sets are logically prior to numbers. The answer to the OP's most recent question is that there is no set of all sets, as demonstrated in the link I provided to a recent thread on that subject. It is true that we may conceptually form the collection of all sets; but this collection is not a set and is not subject to the rules that apply to sets.

Your ideas are a little tangled up, it would be more confusing than enlightening to reply to your post line by line. Instead, give this a read and see if it helps. http://www.scienceforums.net/topic/92101-what-is-the-minimum-number-of-properties-posessed-by-members-of-a-set/?p=900416 All Wiki is saying is that you can't define a set by a predicate alone. Otherwise we'll just use the predicate [math]x \notin x[/math] and derive a contradiction. Rather, you have to start with some existing set, and then apply a predicate to restrict the elements of that set. So in fact we CAN form the set of all real numbers that are not members of themselves. That's just the set of real numbers. No contradiction! (Convince yourself that what I said is true?). The empty set is characterized by the predicate [math]x \neq x[/math]. The empty set is a subset of every set, as you should prove for yourself from the definition in order to clarify your understanding. Grams have nothing at all to do with any of this. There are no sets in the real world. Sets are a purely formal exercise in abstract math. A collection of five apples is not a set in ZFC. ZFC only contains pure sets; that is, sets whose elements are themselves sets. I don't know what you mean by "quantifiable property," but I think you are kind of making that up to try to understand predicates. A predicate is just a sentence with a free variable that has a truth value when the free variable is bound to a constant. In other words if T(x) means "x is tall," then T(Lebron James) is true and T(Tom Thumb) is false. In ZFC you can NOT form the set of all tall things. You CAN form the set of all tall things in some already existing set. (Of course I'm stretching a point here since I just said things in the real world aren't sets.) The point is simply that you can't form a set from a predicate alone; you have to start from an existing set. There is no such thing. The empty set is a subset of every set.

I may have misunderstood. More generally, what Cantor showed was that if we believe in an infinite set at all, we have to believe in different types of infinite sets. Not just the cardinals, but also the ordinals, which are less widely known but equally important. But why do we believe in the mathematical existence of infinite sets at all? They're an arbitrary assumption taken more for convenience. The leap is going from 1, 2, 3, ... which continues without end; to the idea that we can form the completed set of all of them taken at once. It would be logically consistent to deny the mathematical existence of an infinite set. That's an ontological point not always appreciated. There is nothing necessary about infinite sets. They have only fictional existence in our minds, as far as we know. If tomorrow morning the physicists discover an infinite set, that would be different. Till then, infinite sets have the same ontological status as Star Wars characters. Interesting fictions. Now the interesting question (to me) is: Why do our rational and presumably finite minds so readily conceive of the infinite? Why are we able to reason so precisely about something that does not exist in our world? In short, what is Cantor's work telling us about our own minds and about the world?

How does Cantor's argument show that infinity exists?

Incredible. 2011 called, it wants its post back. http://www.scienceforums.net/topic/54219-even-number-odd-number/ See post #6.

There's another famous totally accurate but completely useless test for primality. Wilson's theorem says that [math]n[/math] is prime if and only if [math](n - 1)! \equiv -1\pmod n[/math]. The problem is that computing [math](n - 1)![/math] is even worse computationally than doing trial division. There's actually some theoretical interest to this because Wilson's theorem has other uses in number theory. https://en.wikipedia.org/wiki/Wilson's_theorem

You are taking this way too seriously. Thank you for the explanation. I'll get to work on it. It's a set, of course. A manifold is a topological space; and a topological space is a set. And the real numbers are a set. Hence the collection of functions from a manifold to the reals forms a set within ZF. The context of Xerxes's discussion is Category theory. In Category theory we want to consider all the possible instances of a given type of object. For example, the collection of all the vector spaces over some given field [math]\mathbb{F}[/math]. The problem is that within set theory, there is no "set of all vector spaces" for pretty much the same reason there's no set of all sets. What shall we do? In Category theory we simply assume that we have the entire collection but that it's not necessarily a set. So Xerxes is using the word "ensemble" to avoid having to deal with this point. It's sufficient to realize that the collection of all vector spaces over some field is a collection that's not a set. I would say that "ensemble" is actually NOT a good word to use, since it is the word French mathematicians use to mean set! So an ensemble really is a set, hence using that word in this context is a little inaccurate. Now a question arises: What happens if we want to make sure our categorical reasoning is legitimate? At some point we might want to frame Category theory within set theory, for example so that we can apply set-theoretical manipulations to our categories. There has been a considerable amount of work done in this direction. The usual fix is to place Category theory within a Grothendieck universe, which is an extension of ZFC that includes an extra axiom, one that posits a particular universe that has enough sets to do Category theory. In particular, the set-theoretical universe must contain an inaccessible cardinal, a transfinite number so big it can't be proven to exist in ZFC. Once again we see the principle than math is not based on the axioms; rather, our choice of axioms is based on what math we want to do! But really, that's more than anybody needs to care about. The collection of vector spaces over a given field is a collection that's not a set. That's all anyone cares about in practice.