Only if you are confused. But most mathematicians aren't.
The Countable (Dedekind) Reals
Re: The Countable (Dedekind) Reals
Which are the "non-confused" Mathematicians exactly?jayjacobus wrote: ↑Mon May 16, 2022 5:49 pm Only if you are confused. But most mathematicians aren't.
Is it the ones saying ℝ is countable; or the ones saying ℝ is not countable?
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Re: The Countable (Dedekind) Reals
The ones who count in discrete steps rather than trying to count in endlessly smaller amounts.Skepdick wrote: ↑Mon May 16, 2022 5:52 pmWhich are the "non-confused" Mathematicians exactly?jayjacobus wrote: ↑Mon May 16, 2022 5:49 pm Only if you are confused. But most mathematicians aren't.
Is it the ones saying ℝ is countable; or the ones saying ℝ is not countable?
You can keep trying to count the smaller and smaller divisions but you wil get stuck doing that.
Last edited by jayjacobus on Mon May 16, 2022 6:05 pm, edited 1 time in total.
Re: The Countable (Dedekind) Reals
Which mathematicians are the ones "who count in discrete steps rather than trying to count in continunuosly smaller amounts"?jayjacobus wrote: ↑Mon May 16, 2022 6:02 pm The ones who count in discrete steps rather than trying to count in continunuosly smaller amounts.
Is it the ones saying ℝ is countable; or the ones saying ℝ is not countable?
Which mathematicians are the ones who keep trying to count the smaller and smaller divisions?jayjacobus wrote: ↑Mon May 16, 2022 6:02 pm You can keep trying to count the smaller and smaller divisions but you wil get stuck doing that.
Which mathematicians are going to get stuck doing that?
Is it the ones saying ℝ is countable; or the ones saying ℝ is not countable?
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Re: The Countable (Dedekind) Reals
None would get stuck. They know that irrational numbers are never ending.Skepdick wrote: ↑Mon May 16, 2022 6:03 pmWhich mathematicians are the ones "who count in discrete steps rather than trying to count in continunuosly smaller amounts"?jayjacobus wrote: ↑Mon May 16, 2022 6:02 pm The ones who count in discrete steps rather than trying to count in continunuosly smaller amounts.
Is it the ones saying ℝ is countable; or the ones saying ℝ is not countable?
Which mathematicians are the ones who keep trying to count the smaller and smaller divisions?jayjacobus wrote: ↑Mon May 16, 2022 6:02 pm You can keep trying to count the smaller and smaller divisions but you wil get stuck doing that.
Which mathematicians are going to get stuck doing that?
Is it the ones saying ℝ is countable; or the ones saying ℝ is not countable?
Re: The Countable (Dedekind) Reals
You continue to provide absolutely zero information for me to be able to make an actual choice.jayjacobus wrote: ↑Mon May 16, 2022 6:12 pm None would get stuck. They know that irrational numbers are never ending.
I am still no closer to being able to decide which proposition is true.
Proposition A: ℝ is countable
Proposition B: ℝ is not countable
A or B. Which one?
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Re: The Countable (Dedekind) Reals
You can't decide. That's ok.Skepdick wrote: ↑Mon May 16, 2022 6:15 pmYou continue to provide absolutely zero information for me to be able to make an actual choice.jayjacobus wrote: ↑Mon May 16, 2022 6:12 pm None would get stuck. They know that irrational numbers are never ending.
I am still no closer to being able to decide which proposition is true.
Proposition A: ℝ is countable
Proposition B: ℝ is not countable
A or B. Which one?
Re: The Countable (Dedekind) Reals
Thanks, Captain obvious.
Obviously I can't decide. But IF Mathematicians claim to accept the axiom of non-contradiction they have to decide!
Re: The Countable (Dedekind) Reals
For interested readers puzzled about the question about the possible countability of the real numbers, the Lowenheim-Skolem theorem says that if a first-order theory has any infinite model at all, it has a model of all infinite cardinalities. This implies (counterintuitively, of course) that there's a countable model of the real numbers. This also shows that countability is a relative notion, and not absolute. That is, it's a property that varies depending on the model of the axioms.
There is no violation of the law of excluded middle, because truth is always relative to a particular model. It's like considering the proposition, "There exists a number x such that 2x = 5." This statement is true in the real numbers, but false in the integers. The OP seems to think this is a big mystery; but it's perfectly clear that there is no contradiction. There are simply different models of number systems in which statements that are true in one model may be false in another.
[To be accurate, this is a simplified analogy rather than an example, because the first-order theory of the reals is different than the first-order theory of the integers. In the case of countability, we have the situation where two models of the same theory are non-isomorphic].
The Lowenheim-Skolem theorem was published in 1915, predating intuitionist and constructivist math by years. The OP, who for the record is far too obnoxious for me to interact with anymore, clearly never heard of this and doesn't know this.
Despite his ignorant name-checking of category theory, he used the phrase "universal property" to refer to countability. For people who know some category theory, it's perfectly clear that OP once again is using buzzwords that he clearly hasn't the slightest knowledge of. That's not what a universal property is.
Normally I wouldn't bother to invite yet another ignorant and insulting rant from the OP, but it's possible that there are readers who wonder if the OP has any idea what he's talking about. He does not.
There is no violation of the law of excluded middle, because truth is always relative to a particular model. It's like considering the proposition, "There exists a number x such that 2x = 5." This statement is true in the real numbers, but false in the integers. The OP seems to think this is a big mystery; but it's perfectly clear that there is no contradiction. There are simply different models of number systems in which statements that are true in one model may be false in another.
[To be accurate, this is a simplified analogy rather than an example, because the first-order theory of the reals is different than the first-order theory of the integers. In the case of countability, we have the situation where two models of the same theory are non-isomorphic].
The Lowenheim-Skolem theorem was published in 1915, predating intuitionist and constructivist math by years. The OP, who for the record is far too obnoxious for me to interact with anymore, clearly never heard of this and doesn't know this.
Despite his ignorant name-checking of category theory, he used the phrase "universal property" to refer to countability. For people who know some category theory, it's perfectly clear that OP once again is using buzzwords that he clearly hasn't the slightest knowledge of. That's not what a universal property is.
Normally I wouldn't bother to invite yet another ignorant and insulting rant from the OP, but it's possible that there are readers who wonder if the OP has any idea what he's talking about. He does not.
Re: The Countable (Dedekind) Reals
Oh look at you backtracking in a hurry! Just last week you were defending an ontological view of mathematics, now you have (conveniently) started talking about models.
In the ontological view of Mathematics you aren't working with models of ℝ. You are working with ℝ, right?
Or have you confused yourself between ontology and epistemology.
In the ontological view either countable(ℝ) is true, or countable(ℝ) is false. Which one is it?
If you wish to inform us that you are attempting to straddle the fence on ontology and epistemology - please do so.
Clearly this idiot is lying about the OP has and hasn't heard of. As the search function trivially demonstrates.
Evidently neither does this clown.
One day he's talking about ontology. Another day he's talking about epistemology.
If he knew what he was talking about he would have made up his mind on what Mathematics is about. Maybe he's one of the confused mathematicians who don't really know what Mathematics even is.
Last edited by Skepdick on Tue May 17, 2022 1:27 am, edited 2 times in total.
Re: The Countable (Dedekind) Reals
You are embarrassing yourself.wtf wrote: ↑Mon May 16, 2022 7:34 pm Despite his ignorant name-checking of category theory, he used the phrase "universal property" to refer to countability. For people who know some category theory, it's perfectly clear that OP once again is using buzzwords that he clearly hasn't the slightest knowledge of. That's not what a universal property is.
Directly from the first paragraph of the Wikipedia article you are linking us to.
If two mathematical theories are countable then they work in the same way in the general sense that they are (surprise! surprise!)... countable! Which makes countability the universal property of ALL countable theories. Which is why I am calling them "countable theories". Because they share a universal property - countability.Informally, it [the universal property] represents an intuition that two mathematical theories work in the same way in some general sense, regardless of their specific differences.
That's why it's called intuitionism. I's (supposed to be) intuitive. Not sure where you took a wrong turn because all I am talking about is this.
Code: Select all
{ x for x in theories where countable(x) }
Re: The Countable (Dedekind) Reals
Forgive me if I am missing something among all of this mutual recrimination, but isn't the onus on Skepdick to disprove Cantor's argument for uncountability of the real numbers?
Re: The Countable (Dedekind) Reals
You can be forgiven for being confused. It's a subtlety indeed.
When Cantor and Dedekind are both talking about ℝ you might think they are talking about the same thing.
But if Cantor's ℝ is uncountable and Dedekind's ℝ is countable then... are they even talking about the same ℝ?
The answer is obvious! They are talking about different things! Whose ℝ is the real ℝ? The answer is.... NEITHER!
Mathematicians are never really talking about the continuum. They are talking about finite sequences of symbols that talk about continuums.
There is no THE real numbers.
Re: The Countable (Dedekind) Reals
"The Real numbers are countable (with the fine print being if you define everything just the right way)"
Well, I suppose anything is countable if you define it the right way. Where do you think Cantor went wrong?
Well, I suppose anything is countable if you define it the right way. Where do you think Cantor went wrong?
Re: The Countable (Dedekind) Reals
In your judgement, where does Cantor's Diagonal Argument go wrong?Skepdick wrote: ↑Mon May 16, 2022 9:16 amThe summary of the argument is in the subject line. The Real numbers are countable (with the fine print being if you define everything just the right way).
The moral of the story is that Mathematics is invented, not discovered.
Because it can't be true that both the Reals have the universal property of countability, and the reals lack the universal property of countability.
The "universal property" (what a bullshit phrase) of countability in the Reals is subject to your choice of topos.