No. I was giving practical advice to not overlook that when we read natural language renderings of formulas, then we can't expect that how we naturall...
I don't begrudge anyone from that notion, but, for me, it's too vague. What is "too big"? And which universe? I might suggest saying (only a subtle di...
Hard to discuss a counterfactual here. So let's turn to TF. It's not a matter of being consistent with the axiom schema of specification. Instead, in ...
Read further down in the Wikipedia article, and you will see the axioms for first order PA. There is no predicate 'is a number'. / Regarding your noti...
Moreover, indeed Gitman, just as the Wikipedia article, uses the method of adding a constant as a step in the proof of existence, but then she says ex...
I didn’t say there is anything wrong with Gitman. You seemed to have skipped what I said about it. Actually, I should be sharper by saying that it app...
I should add that study of non-standard models of PA usually considers not merely that the models are not isomorphic to the standard model but also th...
A non-standard model of PA I guess can be visualized through the method Gitman mentions, but literally a model of PA is itself a model for the languag...
No, you have it very wrong. The universe has additional members. We don't add symbols that engender sentences not already in the theory. You are compl...
I don't opine whether they are fictions or not. But, for me, at least such views as fictionalism and instrumentalism that allow sets as consistent fic...
I don't opine on that particular philosophical position. But the outcome of it doesn't preclude existential quantifcation in mathematics or working wi...
That raises the question, "What do you mean by 'the real world'"? And what do you mean by "something exists in the real world"? Anyway, whatever the a...
You want mathematics not to claim that there exist infinite sets; you want to "reject existence", as I understood you, such as existence asserted with...
Of course, we can hold that there do not exist infinite sets. But then providing a formal axiomatization for the mathematics for the sciences gets a l...
In set theory (since you mentioned Cantor, who provided the main pre-formal concepts) there is not an entity called 'infinity' (distinct from a other ...
Adding a symbol is not relevant, not due to whatever you said about sizes of models, but rather more simply that it is not even involved in the notion...
Very much not merely a lack of formality. It is dreadful confusion and misinformation. I try to write correct statements, as best I can within the lim...
Your improvised public "figuring out", with confusions conflating terminology from two different areas of stusy, results in posts that are misinformat...
This is a notion of non-standard model not in model theory (in mathematical logic) but in computer science? It is clearly not the notion in model theo...
The headline reads onto itself and is incorrect. You do somewhat correct it in your post (though, even there, you mistate by saying "ZF minus infinity...
Then you're much better saying it rather than depending a very confused "context". If your usage is is indeed standard computer science, then, yes, th...
Even the title of the thread you posted is wildly, clearly and egregiously incorrect: "You can do with numbers everything that you can do with sets, a...
Yes, it exactly supports what I said. The theory in question is not ZF minus Inf; rather it is ZF minus infinity plus the negation of infinity. In sec...
Unless I'm overlooking something, for any theory, and for any cardinality, there is a model of the theory such that the universe of the model has a me...
Some authors use 'ZF-Inf' to really mean ZF with infinity replaced by the negation of infinity. But better practice is to notate as '(ZF-Inf)+~Inf' to...
alcontali: You are egregiously posting a lot of misinformation Whatever you read on a forum, it's not ZF minus infinity that is, in a certain sense (a...
As I mentioned before, the predicate 'is infinite' is defined as 'is not finite'. You then objected that that is "an illegal move". I explained that d...
I suggest first establishing a firm understanding of the axioms and rules of inference of mathematical proof and definitions - whether in formal first...
Below is a link to an article about HF. (They do call it ZF-Inf, contrary to some others, including me, who call it (ZF-Inf)+~Inf): https://projecteuc...
That is not correct. ZF-Inf is ZF but without the axiom of infinity. (The '-' here means 'without'; it doesn't mean 'the negation of'.) (ZF-Inf)+~Inf ...
Eugene Wright, the bass player with the Dave Brubeck quartet during the '60s, was not white. Perhaps through at least the '60s, whites were somewhat a...
Perhaps one might get the notion of the universe for PA as "a proper class as far as PA is concerned" from the notion regarding set theory that for ev...
I'm not advanced. But I do have a methodical understanding of some basics. (1) There is a difference between ZF-Inf and (ZF-Inf)+~Inf. I'll call the l...
I guess you're referring to something you wrote previously in this thread. Whatever you have in mind, it does not contradict that definitions are not ...
It can't be an "illegal move" since it is merely a definition. The best explantion (an exellent one) I have found of how mathematical definitions work...
There are many senses in which mathematicians use the term 'infinity'. But when we get to formal definitions, we distinguish among those senses. These...
Maybe this will help: PA is a certain set of sentences. There is no sentence in PA that one would ordinarily regard as saying "N is a proper class". T...
I paraphrased what you said: "via the Peano axioms, in which N is a collection but not a set" "in PA, the natural numbers are a proper class" Those ar...
No, by definition, x is uncountable if and only if x is not countable. That's simply a definition of 'uncountable'. Then we prove that there do indeed...
(1) I don't know in what exact mathematical formulation one says "N is a proper class as far as PA is concerned". First order (I'll mean first order t...
There are too many incorrect claims in this thread (and forum) to reply to them all. But I'll address one about the axiom of extensionality. The axiom...
That's not what 'countable' means. Here are the definitions: S is countable iff (S is 1-1 with a natural number or S is 1-1 with N) S is denumerable i...
This is more a philosophical or psychological question than a purely mathematical one, but I don't have much problem understanding that the set of nat...
As I understand, you're asking: What is cardinality of the set of finite sequences on a set of cardinality x? I'm very rusty in math, so let me see if...
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