No, P and N are sets, and f(x) is a function between them, i.e. x is an element of P, and f(x) is an element of N.
((\forall x \in P, \exists y \in N: f(x)=y)
\land
(\forall y \in N, \exists x \in P: f(x)=y))
This is true for the sets P and N as you have described them, and for the (f(x)) I provided.
Ok.
That is your presumption. Prove it to be true for all x and y throughout infinity.
I already gave you my proof for why it isn’t true.
So make it good.
N is the set of natural numbers (excluding zero).
P is the set of natural numbers (excluding zero) and the element A.
First we show that all elements of P are mapped to an element of N by (f(x)):
if (x=A), (f(x)=1 \in N)
if (x \neq A), then (x) is a natural number,
for any natural number (x), (x+1) is a natural number,
so (x \in P, f(x) \in N)
Next we show that all elements of N are mapped to by (f(x)) for some (x) in P:
if (x \in N), then (x) is a natural number.
for any natural number (x) except for 1 (since we’re excluding 0), (x-1) is a natural number.
so for any (x \in N) (except 1), ( \exists y \in P: f(y)=x)
and (f(A) = 1), so for (x = 1), (\exists y \in P: f(y)=x)
so (x \in N, \exists y \in P: f(y)=x)
Therefore, ((\forall x \in P, \exists y \in N: f(x)=y) , \land , (\forall y \in N, \exists x \in P: f(x)=y))
Not really, no.
P is a set of labels for all of the natural numbers plus another label for the letter “A”.
N is a set of labels for only all of the natural numbers.
Neither set is numerical, so your “x” can only be an index, not the element.
That doesn’t show what you intend. All it shows is that you don’t know what happens “at infinity” because your scenario can never go there. In effect, you are presuming the consequent (again). You are assuming that your map is complete merely because every n has an n+1. But your map has a loose, unknown, dangling “unend”. It can’t prove anything about infinity because you don’t know what is going on throughout infinity. It tells you nothing about the size.
Hi James,
Thanks for your post.
As you might assume, mathematics is a large part of how I define myself, and once I start to think that I might be losing my memory in this area it is very disconcerting.
Thanks again Ed
Hi wtf,
I will start with the axiom of existence.
In the text book “Introduction To Set Theory” Third Edition, Revised and Expanded by Karel Hrbacek and Thomas Jech, the first axiom is The Axiom of Existence. This axiom states “There exists a set which has no elements”. Here I will use the notation φ for the empty set, though it is sometimes denoted as { }.
This is very important because 0 is defined as φ. 0 =φ. 1 = {φ}. 2 = {φ, {φ}} and 3 = {φ, {φ}, {φ, {φ}}}. In general n = {0, 1, 2, … n-1}. Roughly speaking numbers are sets of sets of sets … of nothing. I expect that Shakespeare might think that this is “Much ado about nothing”. You can find this axiom, it is the second axiom on ZFC, in the online Encyclopedia of Mathematics.
I think that Wiki realizes the problem with this axiom and goes out of its way to define the empty set in terms of other (hopefully) existing sets. This relisting of ZFC axioms (which should be independent, but in fact are not) ultimately makes the Natural numbers lose its uniqueness.
My opinion: Over 80% (some Algebraists use their own foundation) of all mathematicians would claim that ZFC is the foundation for mathematics. Yet, maybe 95% of all mathematicians have no idea of what ZFC is. There are extremely few schools in the country that teach foundational mathematics.
Bijections:
A bijection is a mapping from Set A to Set B such that, for every y in B, y = f(x) for some x in A. Furthermore f is a unique mapping. Formally f(a) = f(b) if and only if a = b.
These types of maps are very important in Cantor Theorems, they form the way that we measure the size (Cardinality) of various sets.
Examples of sets that have the same Cardinality are the Counting numbers (The Set = {1, 2, 3, …} and the Set ={0,1,2,…}. The Set = {1, 2, 3, …} and the even numbers. The Set = {1, 2, 3, …} and the odd numbers. The Set = {1, 2, 3, …} and the Cartesian product of the Counting numbers CXC. The Set = {1, 2, 3, …} and the Integrers. The Set = {1, 2, 3, …} and the Rational numbers and even The Set = {1, 2, 3, …} and the Rationals X Rationals X Rationals … . Two sets do not have the same Cardinallity are the Counting numbers and the Real numbers.
Homomorphisms:
A homomorphism is a bijection map from Set A to Set B such that, if x (think generalized multiplication) is a binary operation on Set A and there exists x’ ia binary operation on Set B, then f(a x b) = f(a) x’ f(b). Note x and x’ can be generalized to n-ary operations and the property that then f(a x b) = f(a) x’ f(b) does not need to be restricted to single operations.
This means that with respect to the operations x and x’ the sets A and B mimic each other. I used this type of mapping, in one of my posts, to show how some aspects of the empirical world can mimic the Natural numbers and vice versa.
On 0.000 … :
I think that this is at least an ambiguous expression and likely to have additional interpretations.
If we assume that the expression is a function f then we could write that f is a function from the set of Natural numbers to the set of Counting numbers given by the map (n, f(n)) where f(n) is defined as follows:
f(n) = 1 + 0(∑ from i =0 to i=n of (1/10)^i).
Since 0 times anything is 0, f(n) = 1. However, and this is where we differ, f(n) does not denote f. It is simply the y coordinate in the set of ordered pairs {(n, 1)} which describes f’s mapping.
A simple observation:
It is important that the indices in the summation be precise. E.g.:
∑ from n = a (where a is a Natural number) to n = ∞ of Exp(n) is defined to be a limit and not in itself a function.
However the following summation simply indicates that n is what L.E.J. Brouwer called an indefinite finite number:
∑ from i = a (where a is a Natural number) to i =n of Exp(i).
The above summation forms a proper map. (Assuming that Exp(i) is appropriate).
Thanks Ed
I don’t know whether to laugh or cry. I think I’ll laugh. Your questions are challenging. I need to cover some math ranging from freshman calculus all the way up to grad school.
To this end I’ll be as brief and clear as I possibly can. I’ll hit things at a very high level then provide explanation and detail as requested.
Before starting note that in math you can make up any notation you want. Mostly you’ll be ignored, but you certainly have the right. And if you happen to solve a problem or find a new connection between different areas, you’ll become famous and everyone will adopt your notation and concepts. Or more likely someone else will get credit. That’s how it usually works.
Math is not static. Math evolves as a sequence of new ideas and profound surprises.
Ok enough philosophy. Let’s “do the math.”
In calculus we want to be able to talk about expressions like (\displaystyle \lim_{x \to \infty} f(x)) or (\displaystyle \lim_{x \to 0} \frac{1}{x} = \infty). To do this we augment the real numbers with a pair of purely formal symbols (\infty) and (-\infty). We give them some handy formal properties to make our notation work out the way it should, and that’s all they mean. No ontology, no metaphysics. They’re not the Alephs or the hyperreals or anything else. They’re just formal symbols used as a notational convenience. Any “meaning” is a matter of one’s personal intuition. .
I’d say that a lot of people wonder or think that the infinities in the extended reals have something to do with the Alephs, and they just don’t. Math has multiple ways of formalizing and dealing with infinite; and these various concepts are not necessarily related.
The concept you want here is ordinal numbers. These are the transfinite numbers that express the concept of well-ordering. A set is well-ordered if every nonempty subset has a smallest element.
For example:
The natural numbers are well-ordered as (1, 2, 3, \dots).
The integers are NOT well-ordered because, for example, the negative integers (\dots, -3, -2, -1) are a nonempty subset without a smallest element.
However we can REORDER the integers so that they ARE well-ordered, as (0, 1, -1, 2, -2, 3,-3,\dots). So we see that given a particular set containing some elements, we can regard its elements as being well-ordered in various different way.
The ordinal numbers encapsulate the concept of a well-order. They are very strange and interesting. They are also logically prior to cardinals, since in the modern formalism we define cardinals as particular ordinals. [This replaces the classical notion of a cardinal as the collection of all sets equinumerous to a given one. The problem is that such a collection is not a set, so we abandoned that approach].
Now the ordinal corresponding to the cardinal (\aleph_0) is called (\omega), lower-case Greek letter omega.
So if you wrote (\displaystyle \lim_{n \to \omega}), my guess would be that you are doing set theory involving ordinals. However there’s no context in which this notation represents anything interesting, and it would be unusual to see it. And it would be a really weird thing to see in calculus. It would be totally out of place.
[Warning, Grad school math ahead. But it’s not that hard].
Now think about what (\omega_1) is. It represents one way (among many) of well-ordering the elements of the cardinal (\aleph_1). So (\omega_1) is in bijective correspondence with (\aleph_1). In other words, (\omega_1) is an uncountable set.
But on the other hand, it’s an ordinal. So by definition it’s a well-ordered set. (\omega_1) is an uncountable ordinal. It’s the smallest uncountable ordinal there is. That’s why it’s known as the “first uncountable ordinal.”
You cannot conceive of such a thing. If you CAN conceive of such a thing, that shows you don’t understand it! Because we literally cannot imagine an uncountable ordinal.
We can prove that (\omega_1) exists in set theory. In fact we don’t need the axiom of choice, so the mathematical ontology of (\omega_1) is quite solid.
So when we write (\displaystyle \lim_{n \to \omega_1}) that does not mean anything at all unless we define what we mean by it. Since sequences are defined as functions whose domain is the natural numbers, then sequences are are “too short” to reach (\omega_1) as a limit. Sequences are only countably many elements long. We can’t use them to define limits of uncountable ordinals. Instead there is a more general concept, called nets.
I hope some of this is helpful and/or interesting.
Oh you mean the axiom of the empty set. I should have realized that’s what you meant.
I’ll respond to the rest of your post tomorrow. The axiom of the empty set is not actually agreed on by everyone. For example we can take a version of the axiom of infinity that says: “There exists an inductive set.” And since that says that SOME set exists, call it (X), we can apply the axiom of specification to form the set (\emptyset = {x \in X : x \neq x}) to instantiate the empty set. When you do it this way you kill two birds with one stone, since you get the natural numbers (the smallest inductive set) and the empty set in one axiom.
With the empty set axiom approach you need TWO axioms, since you have to say that the empty set exists, AND an inductive set exists that contains the empty set.
It’s a matter of preference and of little consequence.
But I’ll certainly agree with you that we have to stipulate that SOME set exists, else for all we know we could write down all the axioms of set theory but there aren’t any sets! So we’d be reasoning about purple dinosaurs, things that don’t exist.
I hope you’re not referring to the Ra notation. That’s of no consequence whatever. It’s likely that it’s an obscure notation used by some old book you ran across. Or perhaps it’s European or Chinese or something. I have no doubt you’ve seen it somewhere. I hope you don’t think anyone’s making an issue of this.
Brouwer was a bit of a foundational mystic (after starting his career in “hard” math) and didn’t believe in the standard real numbers as they’re commonly understood. He founded the mathematical philosophy of intuitionism.
IMO we should steer well clear of Brouwer. His ideas about the continuum are very murky. Of course we can discuss intuitionism but I don’t want to confuse it with the standard real numbers.
In fact, all that tells us is that both sets are infinite. It says nothing at all about their relative size. Is one a greater degree of infinite than the other?
We agree that both are infinite and thus there is always an “n+1” for every n. We have to know more than that.
When we start with two identical sets;
[list]P = [1,2,3…]
N = [1,2,3…][/list:u]
we immediately know that they are the exact same size and degree of infinite … because they are identical.
When we add another element to one of them, regardless of what size they had been, by definition, that one set is 1 element greater than the other.
[list]P = [A,1,2,3…]
N = [1,2,3…]
P = N + 1 element. <<-- relative size[/list:u]
This is a distinction without a difference. We can refer uniquely to elements of the sets by their indexes, and we can uniquely map from P to N by reference to their elements’ indexes.
I haven’t relied on the notion of anything happening ‘at infinity’. I’ve relied only on properties of the natural numbers.
It’s a property of the natural numbers that each has a successor, i.e. if (x) is a natural number, so is (x+1). And for every natural number except 0 (or 1 in this case), the inverse is also true: if (x) is a natural number (>1), (x-1) is also a natural number.
Do you reject this property of the natural numbers? If it’s true for all natural numbers, my bijection works.
It is. I should have anticipated that the answer would take me out of my depth. But nets are new concept for me, so I am glad I asked.
Not always.
Oh, you certainly have.
The proposal is that one set is a greater size than the other. Without knowing what is happening “at the end” (as it were), you can’t know if one element has been left out of your scheme. And in fact, as I showed, you do leave one out.
I have utilized that property too. But unlike you, I didn’t try to push the additional element off of the non-end where it never gets counted. All you proved is that both sets are infinite. We knew that.
Your bijection fails, as per:
On the other hand, you have not made any relative size comparison at all.
So you cannot claim that your bijection worked. And as proven above, the bijection is false.
When two things are identical and we add to one, we have said that one is everything that there other is plus more. So by definition, we already know that one set is greater than the other, else we didn’t really add anything.
Which element cannot be uniquely identified by reference to its index?
The proposal is that there is a bijection. The concept of size appears neither in your proof nor in your subsequent change nor in my proof. My proof proves only that we can define a function that, for every element of P, maps it to a unique element of N, and that every element of N is so mapped by the function.
We were talking about relative size and someone proposed bijection as a measure. And I ask for you to prove that bijection functions properly for infinite sets. You did not show that.
What you showed was that if we shuttle any additions off to the end of the natural numbers, they will never be counted in the bijection. After the bijection of all of the natural numbers, one set still has one element unpaired. Your scheme doesn’t account for that. Mine does.
I mentioned nets only to show that sequences aren’t “long enough” to define limits in uncountable ordinals. So that the notation (\displaystyle \lim_{n \to \omega_1}) is not possible to make sense of without introducing more sophisticated math. I do hope I was able to shed light on your notational questions. When we talk about infinity in calculus we are only talking about the extended reals, which have nothing to do with the Alephs.
That said, if I explained an uncountable ordinal and you didn’t say, “Wow, I had no idea such a counterintuitive thing could exist,” then my exposition totally failed.
Every natural number has a successor, so that element can be paired with it’s successor.
Definitely. It’s useful to be reminded that these notations have more rigorous definitions, and that they can be abused pretty easily by improper uses not contemplated in the underlying definitions. It similarly occurs to me that I’m conflating two meanings of limit (i.e.in the case of a function, and in the case of a sequence) that while intuitively similar, need to be formally defined in different ways. A limit using natural numbers is meaningless for a function (\mathbb{R} \to \mathbb{R} ), so it seems expected that (lim_{x \to \omega_1} ) doesn’t make sense for that function either.
Conflating two meanings of limit, is at the essence, but what does their difference consist of? Is there a single function which can describe it? Or, in what sense can such function be primarily an intuitive pre-positional function consisting of the supposed sum of all differentials?
This question has the feel of a dog chasing its own tail, but does a resulting picture of a tautology come in mind, where, after so many runs, wearily, the admission of circularity comes in?
Granted at this point .999999999 may result an tentative equivalency to 1, where it could be claimed, that there may never ever be a point where a differential between the two types of limit could not become infinitesimally significant. But that is not to say, that such insignificance does not ever show a language whereby the absolute identity can be verified.
And finally, is there a difference between the insignificance of the former, and the possible significance of the later? And here ,the admission has to be yes,but by only an insignificant value.
If significance is measured by transcendental values, then then one set of variables would be used, in the other case,another. Meanwhile as the function nears the immeasurable infinity, there may be an effect close to what is described in cosmic measurement, as shifts , is n qualitative terms, whereas here may occur an expansion of perceived quantified values, setting up new signifiers, but occupying an equal spatial volume. This process as well can go on until newer and newer limits are reached.
But infinity as a concept breaks down definitionalky, since any conception, including infinity, confines it with its nemesis, its non-infinite counterpart. By this token, infinity and its finite counterpart, limit each other by the same limit: and hence set up multiple bubbles of limited infinite universes.
The reason this does not work, is, or may not work, is, that such a conception fails to account for repetitive singular universes, and mistake it for identical but different ones.
There is no way a repetition of two separate spheres can be differentiated from the Omni presence of one sphere identical to its self.
The problem of a cosmological and quantum integration has to take place, before it can be de-integrated and understood each within its own sense.
So until then the two propositions are at once different, and the same.
Mathematics need not dwell on it until that time where it will become significant to do so. It may ,when ultra large accelerations will be needed to attain intergalactic velocities, when relative velocity transformers can integrate the temporal specification of a cyborg, for instance, to correspond it with say thousands or millions of light years.
At the rate that science doubles every current bounded time, this is not at all as far fetched as it seems. But will the tempo of natural time, as corresponded with natural numbers allow this, or such stretch even stretch the imagination of the universe?
Sorry into the venture of the imagination into fictional territory, but was compelled, on account of this IF within the compass of available technological need for near infinite levels of quantification.
will the human animal animal be capable, or even deemed deserving for such stretch in space time, or even in imagination, lest they fall coming too near the ultimate source of energy?
This is exactly the point on which there is confusion.
I claim there should be no confusion and there is no confusion.
The word limit is a term of art. It is “a word or phrase that has a specific or precise meaning within a given discipline or field and might have a different meaning in common usage”
Term of art is itself a term of art in the legal profession. In the law they always have cases involving highly technical fields. They understand that each field has its own technical jargon. And that this jargon must be taken on its own terms, exactly as it is used in the technical discipline in question; and the meaning is NOT to be taken as having relation to the everyday meaning of the word or phrase.
If someone says, “Well, a limit is really not the same as the way mathematicians define it.” the answer is that “Yes it is!” Because the word limit is a term of art in mathematics. When we are doing math, the word limit has no other meaning other than the formal one. When a physicist talks about force or energy or mass or inertia, those terms have extremely specific technical meanings that are only tangentially related to their use in everyday speech.
Why are people holding mathematical terms of art to some requirement of conforming to what random people on the street think the word means? In math limit doesn’t mean anything other than what mathematicians say it means. Just like physicists agree on what force and mass and acceleration are. They are very specific technical things. Not everyday things. They’re terms of art.
As James tends to say … priesthood.
Only the anointed ones understand. Only they truly know.
The peons need only repeat what they are told.
Right, James?
James is right. Physics is a priesthood. The law is a priesthood. Medicine is a priesthood.
If you walk into a room and flip on the light switch, you are leveraging the billions of dollars of technical infrastructure and decades of technology it took to pump electrons to that lightbulb. What makes you think everyone at the power company knows what they’re doing and have your best interests at heart? In modern society we have no choice. We’re forced to take the word of experts.
Is that what this is about? Authority? If so, I’d rather trust someone who tells me that (.999 \dots= 1) than someone who tells me they know better than I do how I should live my life. If you don’t like authority figures, push back on the politicians. The mathematicians are harmless.
Personal note. I lived two miles from the San Bruno pipeline explosion. Your time would be far better spent investigating the corruption and incompetence at your local gas and electric company than worrying that mathematicians are lying to you about limits.
No, because “A” is not one of the natural numbers being paired. You only pair the natural numbers, not the one left over.
At this point, you know that your “proof” falls short and that you have no argument against my proof. Of course, you will continue to deny everything and repeat your argument forever, as always. So no need to continue bantering with you.
As is always the case, it seems, you provide proofs with flaws and I provide proofs within which you can find no flaws (two in this case). And at that point, it becomes nothing but repetition.
… So sayth the Lord. [-o<
Math has not escaped religiosity, it seems.