@user21820 Can you elaborate it a bit ? Because I don't quite understand what you meant by "You keep forgetting that the bound is on the absolute value" =)

The inequalities you wrote are wrong because you did not use this. o(1) includes both positive and negative values.

That's why I told you to do this instead. It's not always possible to invert in order to transfer bounds, but here it is possible so might as well.

Thanks for accepting my request to join @user21820

@user726941 Hello, and welcome! Do you have a mathematical inquiry?

nope, just an interested self learner :-)

I did ask a basic math question on the main site that did not get any answer, may i post it here please?

@user726941 Sure.

Thank you 🙏

@user726941 Why do you keep requesting access to the other chat-room? It's clearly unsuitable for you; as the room description clearly states, it is for advanced logic.

sorry

I'm willing to learn?

@user726941 That's what this room is for; read the room description. I've posted a comment on the thread you linked. If you've any questions about that, you can ask here.

ok, will do

@user21820 I have another question regarding constants and variables.

Oh that was fast, I was just writing down my example.

If its too advanced for me to understand, as always, let me know, ok?

But I think its rather basic.

If I have a finite set S, then, by definition, there is a natural number n such that S is bijective (via f) to a set of numbers smaller or equal than n. Then it is common to name the objects of S according to this bijection, meaning s_i:=f(i). Can I always give arbitrary names to objects? Do I need to know that they exist? Are there any other restrictions?

Again, I use name and reference as synonyms.

Because when writing name I always have the picture of an underlying object that is called by that (and possible other) name(s).

I do not understand your question. Things that don't exist are not even objects...

Do you understand the question if you ignore the existence part?

If so, you can also answer that. Maybe the existence part doesnt make sense.

Namely: CAn I always give arbitrary names? Are there any restrictions?

Well, if you have a function from [1..n] to S, it does not mean that [1..n] are references in the sense that variables are. [1..n] are objects and your function also is an object that relates them to members of S.

Yes, those are different in the sense that these references are constants and variables are not.

Right?

No. Variables in the foundational system are always only strings.

Variables are not objects.

Variables refer to objects.

Where did I say something that contradicts that?

If you have a bijection from S to T, the members of S are not references to T. They are just what they are, and it so happens you have a mapping.

I meant to say that s_1 after definition, is a reference to a constant object, namely the object that s_1 refers to.

Whereas a variable n could refer to any object. Only after I write something like "Given n in N" I pin down a specific but unknown n.

But that is not what I meant to ask

@user21820 That is also something that holds for any symbol, doesnt it?

@MaxH Yes. Constant-symbols are not objects, but they refer to objects.

@MaxH "s_i" is meaningless.

Right, this is what I understand then, even though I fear that you don't believe me yet. :P

But again, that is not what I actually meant to ask, even though I struggled with that in the past.

Do you want me to specify what I meant to ask?

Actually, no. It is quite clear that all your questions are due to inability to actually use a foundational system for mathematics. Here, we need ST (set theory), and if you know how to use ST then you know that an indexed family (s[i]) is nothing more than a function, so if you have f∈[1..n]→S, then there is absolutely no point in defining s[i] = f(i) for every i∈[1..n], because it's no different from f!!

Just remember that nothing you define in mathematics are ever names/references to mathematical objects. They are always objects.

@user21820 I don't see how that is contradiction to what I say.

You used "name" and "reference" wrongly.

Correct me if I am wrong, but as far as I know, an indexed family (s_i) is given by a function f:I to S and one then defines s_i to be the object in S that is the image of i under f.

@user21820 I said earlier that I use them both as synonyms. Is there a problem with that? If so I am totally willing to adapt, but I didnt think there is.

@MaxH You're wrong, and I already explained why, but as I said you're not ready to understand it.

You can never define just s[i]. You are actually defining s, which ends up being exactly the same as f.

I dont see how I am wrong, as this is analoguous to what is written on pages such as wikipedia and lecture notes.

So it is utterly pointless and only reveals inability to use a foundational system...

@MaxH So if others do not truly understand the logic needed for what they are doing, you want to follow them? That's fine, but I thought you wanted a full understanding.

No that is not what I said, it is just that it seems weird to me.

Since it appears to be "commonly applied", if that makes sense.

@MaxH As for that, both are wrong. I already said; everything there is an object, never a reference or name or whatever you seem to want to call that notion.

@user21820 Ok so now I am getting lost in quotes. What do you mean with "everything there" where?

Look, if you do not understand that s = f, then you need to stop and realize that your understanding is wrong.

With s you mean the indexed family? Because I think you never explicitly said what s is supposed to be.

Again, the notion of indexed family I used was exactly the one you can find on wikipedia. If that is wrong then what I learnt is wrong.

@MaxH ??? It's the s that you gave. Just answer the question. Is s = f or not?

So you mean capital s? S?

The set?

I never gave a small s

Look, I have said enough times that you do not understand enough. That's why you keep asking again and again the wrong questions.

Yes, but that is something that I never gave, which is why I am confused.

Yes, its fine if you say that I am not ready to understand it yet. I thought it was a simple question with a simple answer, since it is commonly used everywhere.

I'm not going to answer anymore if you're going to insist that you never defined s. That is precisely what your message is essentially doing. The fact that you do not realize that shows clearly that you lack the understanding.

And this isn't something that can be fixed by me answering your questions, because you clearly lack the ability to understand the answer...

So you need to learn the basics first...

@user21820 Which I am totally fine with. I literally never mentioned an s, I gave an indexed family (s_i), a set S and a function f. Anything else that you point to is something that you have to tell me what is meant by. If you say I have defined an s when I only mentioned (s_i), f and S, then I don't know how that is not supposed to cause confusion. I still dont know what you mean with s and if you dont want to explain that I wont know it either.

If you mean anything of those with s, then you have to tell me, because how can I know?

Anyways, if you dont want to comment further, then I am sorry for stealing your time, as that was not my intention.

@MaxH This will be my last comment on the matter. You cannot define just "s[i]". You thought (falsely) that you did. The reality is that you defined s, and the notation "s[i]" means the item given by s and index i. But that s is identical to f, and it is just foolish to think otherwise. You can't see the foolishness until you understand how to use ST.

Well, I know that (s_i)_{i in I} for a set I is indeed the same as f, yes. If you refer to this indexed family as s, then I am not getting at all how I should know that without any sentence of you explicitly stating this. And this is exactly where the problem was, that I didn't know that you refer to the family with s.

@MaxH That is the meaning of the notation you used. You can hardly blame me for you not knowing the meaning of what you wrote...

But if you agree that s = f, then there is no meaning to your question, because there are no names or references anywhere.

I am not meaning to blame you at all. It appeared to be a misunderstanding from the start, which is often the case, reflecting on our chats.

If it appeared to you that I was blaming you, then I apologize.

I am here to learn, nothing else. Blaming anyone doesnt do any good.

Ok, but tell me, where are the so-called "names"?

Maybe this is wrong and at the core of the question: If I have a function f:X to Y and an object x in X, then f(x) is a reference to the object in Y that is assigned to x via f, right?

@MaxH No.

Then this is probably where the problem lies.

I remember making it clear to you that "1+2" ≠ 3.

"1+2" refers to 3.

"f(x)" refers to an object. But f(x) is that object.

You need to be very clear on the difference between a reference (which is a string in the proof you write) and an object (which does not exist in a proof).

So the mistake I made was that I didnt write quotation marks? Meaning, I should have written "f(x)" is a reference to the object in Y (...).

Well yes, this is the mistake in your final question, but I think it is clear that it is not that you forgot quotation marks but that you were actually confused between names/references and what they refer to. After all, if I use your own words, even if you put quotes you were claiming that it is common to name the members of S by using f, and that "f(i)" is the name of f(i). This makes no sense whatsoever.

Given f∈I→S and x∈I:
  f(x)∈S.

@user21820 So I think I now notice something I appear to do wrong. This wont solve the question, though, as I don't really seem to see what the problem in this is. But let me think of how to word it.

"f(x)" is a string with 4 characters. It certainly cannot simultaneously refer to different objects. In the inner context above, it refers to an object, and based on what "f∈I→S" and "x∈I" mean, we know it refers to a member of what "S" refers to. But you cannot say that each member of S has a reference given by f. That makes no sense.

When we say "member of S" when talking about the proof, we of course mean "member of what 'S' refers to".

@user21820 Yes! This is actually how I use it with the thought behind it. The underlying object that "S" refers to.

@MaxH It doesn't matter. What you said earlier makes no sense.

In no way can you ever think that "f(x)" refers to f(x).

@user21820 This is a line that confuses me.

If you don't believe me, attempt to say what you said previously (using just functions, don't confuse yourself further using indexed families) but very precisely; if you mean "what ... refers to", don't just say "...". You will find that you will simply be producing nonsense. Just try.

@user21820 I dont understand whats wrong here. I have an object in Y, namely the object that x is assigned to via f. I want to refer to that object with "f(x)". Only now I know that "f(x)" refers to that object. When we then write f(x) we mean the object in Y that "f(x)" refers to. This is how I thought it is.

@MaxH: I feel like this particular inquiry is a waste of time because you clearly don't get the issues. I'll try one last attempt to explain what is correct, ignoring all the wrong stuff you said above. If you understand what I said, then good. If not, then forget about it. Whatever the case, make sure you finish all the FOL exercises.

["f" and "I" and "S" are strings, and in this global context each of them does not refer to anything.]
Given I,S∈set:
	[In this subcontext, "I" and "S" are variables that refer to sets.]
	[Purely for the sake of discussion, let I' and S' be the sets that "I" and "S" refer to.]
	["f" still does not refer to anything.]
	Given f∈I→S:
		[In this subcontext, "f" refers to some function f' from I' to S'.]
		[f' is NOT a function from references to members of S'.]
		["f" is also still NOT a function.]

@MaxH: There you go. Everything I wrote there should be correct. Remember, "f(x)" is the same string with 4 characters regardless of where it occurs in the proof, and whether or not the proof was written. At some points in the proof, it is meaningless. At other points, we should understand it as referring to some object. But what "f" refers to never assigns names/references to objects, nor maps names/references to objects.

Thanks.