uncool

I may be severely misreading you; it seemed to me that you were implying that for something to be an axiom, it would have to contradict/be inconsistent with what came before it. Which is incoherent. If that's not what you meant, then I don't understand why you asked about whether the assumption of infinity would be inconsistent. Not in model theory. In model theory, approximately (I am not a model theorist specifically), there are theories and models. Theories are systems of axioms; models of a theory are things that satisfy those axioms. When talking about a theory, you can talk about statements of that theory; they may be categorically true, categorically false, or neither. In a model, every syntactically valid statement is either true or false. For example, there is the theory of groups, where the axioms are associativity, identity, and inverse(s). A model of that theory is S3, the symmetric group on 3 letters; another is C3, the 3-element cyclic group. Commutativity is a statement; it is true for C3 but not for S3. The class of hereditarily finite sets is a model of ZF(C) - infinity; this implies that infinity cannot be derived from the other axioms.

Yes, that's my point. What step in your proof fails if you try to apply it to the natural numbers instead of the reals? Where do you use "denseness" in your proof?

If an assumption would otherwise be inconsistent with the other axioms, making it an axiom leads to an inconsistent system. Your concept of an axiom seems flawed to me. The other axioms neither imply nor exclude the existence of an infinite set; the axiom of infinity is independent of the other axioms (assuming the usual assumptions of consistency of ZFC). I am using the word "model" in the precise sense of model theory. The class of (hereditarily) finite sets forms a model for ZFC minus infinity. I don't understand what you mean by "precede" here; I am saying that that class follows the rules of the other axioms.

What do you mean by "be in conflict with"? If you mean that such an assumption should lead to a contradiction, then I think your concept of an axiom is faulty. The closest thing to sensible I can see is if you are asking "What models of the other axioms are excluded by this assumption?" To which the answer is models where all sets are finite. In some sense, to get infinity, you have to start with an infinite set in the first place.

There actually is an axiom which is essentially that the natural numbers exist (more precisely, that we can construct something which follows the Peano axioms): There is a set S such that the empty set is an element of S, and such that if x is an element of S, then {x} is an element of S. This is called the axiom of infinity, and the natural numbers can be constructed from it. (There are other constructions, like "if x is an element of S, then the union of x and {x} is an element of S"; all that matters is that the operation chosen is injective and doesn't send anything to the empty set)

I assume this is the post? The problem is that you seem to have misunderstood the well ordering principle by accidentally rearranging quantifiers. It doesn't say that there is a minimum that works for all sets; it says each set has its own minimum; that is, while multiple sets can have the same minimum, they are not all required to. z may not be a minimum of T, but another thing can be. Your proof uses nothing about the real numbers. If it worked, it would apply equally well to the natural numbers, which are well-ordered. Try it out, and see if you can understand why it doesn't work there. Alternatively, try writing your proof formally - you should be able to see where you have wrongly exchanged quantifiers.

Precisely - which means you have no natural number corresponding to the set of even numbers, which means you have no bijection.

There is a genetic component, however, that says "When you are getting these signals, do this." I think the relevant part may be Hox genes; an example is when scientists were able to cause flies to grow antennae in the place of legs. https://en.m.wikipedia.org/wiki/Antennapedia

The axioms are designed to work together. Just saying "limited by ω" doesn't really work. For the rest of your post, the idea of formal sums covers what you are trying to do. You are trying to deal with the concept of formal sums over an ordered set.

This is not in any way how the usual natural numbers work. I'd recommend using a different name. N' or EN or something like that. So you've defined these new "pseudo-natural numbers". OK. Now what do you plan to do with them? Note that anything you prove with these "pseudo-natural numbers" does not affect statements about actual natural numbers.

I can't figure out what that means, no. You're clearly using "+" to mean something different from how it is usually used. To answer a later question in your post, yes, since you are using "+" in a nonstandard way, I do expect you to define it.

Any finite number? So the string "101010101010" by itself represents the set of all even numbers? Can you type out the specific string you think represents the set of all even numbers? No ellipsis - the exact string.

If you're adding an axiom, you will have to demonstrate that your new system is consistent (or at least, equiconsistent).

You would have to prove there is such an ω. When considering the integers, there is no such ω; when working with extensions, you will have to define your operations (addition, subtraction, etc.) If you are truly usinng cardinalities, then there is no subtraction. ω-1 does not make sense. I'm not "creating them". B does not contain elements that aren't in A. For any integer x, 2x is also an integer.

Again: a bijection doesn't exist "step by step". It exists all at once. For every number, it is possible to double that number. This process is injective, and by definition of even numbers surjective to the even numbers. Therefore, it is a bijection.

N has infinitely many members, each of which has finitely many digits. So there is no natural number "n" that denotes the number of elements of N.

A bijection isn't (generally) a step-by-step thing. It can be built step-by-step, but the function simply is. For example: the identity map on the natural numbers, that takes each number to itself. This can be built step-by-step, sending n to n, but it simply exists: given any n, it returns n. It is clearly a bijection, just by checking definitions.

As strange has said, that is not a natural number. If you think otherwise, then please write that many '1's in a row. For example: I can write 2 1s: 11 I can write 15 1s: 111111111111111

As with all your other attempts, pengkuan, this fails because you exhaust the finite subsets, then claim to have exhausted all subsets. What natural number corresponds to the set of all even natural numbers?

How did you get those series in the first place? They usually would come from calculus, but you are trying to define calculus in the first place. It would look a lot like (be isomorphic to) the field of rational functions on R, with order given by "eventual" behavior/leading coefficient.

Technically, it does. You can define the formal "number" infinity, adjoin it to the real numbers, and use the resulting field. The problem with using this for calculus is that you then have to define your functions to include those new points. What is sin(1/infinity)? What is e^(1/infinity)? Alternatively, can you define the function f such that f(x) = 0 unless x is positive and smaller than any positive real number, in which case f(x) = 1? There are a couple ways around that. One has already been discussed - using nonstandard analysis, in which case the above f doesn't make sense (kind of). None of them are especially useful, especially to undergrads.

I believe you've said you favor the "Dedekind cut". There are two ways the Dedekind cut can be used to define the real numbers: as part of a definition, and as part of a construction. Definition: The real numbers are an ordered field so that for every Dedekind cut (depending on the definition used, you may need to add "nontrivial"), either the lower or upper set has a maximum or minimum, respectively. If you are using this, then the set of real numbers of the form "infinity hat minus b" are larger than anything not of that form, so the two (that set and its complement) form a Dedekind cut. What is the corresponding maximum or minimum? Construction: The real numbers are the set of Dedekind cuts of rational numbers. If younare using this, then please describe the Dedekind cut of rational numbers that gives the "real number" "infinity hat minus 5".

1) Using the standard definition of real numbers, you can't. If you think you have a definition or construction that will allow it, you will have to provide it; in doing so, you may have to demonstrate the properties you usually take for granted. 2) You defined (or claimed) "neighborhood of infinity" to mean numbers that can be expressed as "infinity hat minus a real number". That is not standard at all, and would break the usual real numbers. 3) You are discussing your own construction, which is not the usual complex numbers. No complex number (or real number) is of the form "infinity hat minus b".

Studiot: " Finally he comes at the end to state that if he can find a number, off the critical line, whose zeta function is zero then he has disproved the theorem. That is self evident. But he has to show that (at least) one such number exists." He does show such a "number" "exists" by abusing properties of infinity. sevensixtwo: 1) You cannot "separate" the real numbers into a neighborhood of zero and a neighborhood of infinity, so that each real number lies in exactly one of the two. 2) A neighborhood of infinity is not usually defined in the way you have chosen; the way you have chosen is inconsistent with the usual properties of the real numbers. 3) The Riemann hypothesis is about the complex numbers. If you want to talk about it anywhere else (for example, in this "neighborhood of infinity"), then you are dealing with a problem different from the Riemann hypothesis. 4) If you are going to shoo people off to links about papers that have been discussed here already, you probably should address the criticisms of those papers, too.

"Let say that instead of TMH, we have a TM that simply displays a ''0'' whenever a TM exceeds a fixed number of steps." When you say a fixed number, do you mean that this one TM, say, checks if another TM goes longer than 1000 steps, outputting a 1 if it stops before then and a 0 otherwise?