Your distinction between 'calculus limit' and real number is irrelevant. 1/3 is a number. It has the decimal representation 0.333.
It has no exact decimal representation. You can obviously use decimal notion to approximate 1/3 as close as you like. However since you can never write out an infinite number of 3's or actually use that representation in any real way.
What good does it do to even claim that 0.333... equals 1/3? Other than a useless empty claim it has no practical value. And even in mathematics, when we say that 0.333... = 3 we're actually making this statement using the calculus limit as our justification. So why not just say, "In the calculus limit 0.333... = 3". To do anything less is nothing short of laziness. What you are arguing for is nothing short of lazy mathematics.
What rule of numbers or mathematics says that I cannot add zeroes at the end of a given number?
Obviously it depends on context. If you are claiming to have complete list of numbers for that numerical system then you can't just arbitrarily add zeroes without taking into consideration how that will affect the complete list.
It seems to me, you continue to be confused by the ambiguous way in which you are using the term 'complete,' although I accept you can define it as you clarified below.
I've clarified this several times over already. By 'complete' I simply mean every number that a numerical system can represent.
You seem to be ignoring the context of Cantor's argument. Cantor is trying to claim to be able to produce a number that isn't on the list. Well, if his list isn't a 'complete' list then his claim to have created a number that isn't on his incomplete list is utterly meaningless. He hasn't proven anything if the list he's creating is incomplete.
So that's why it's paramount in this situation and cannot be ignored.
Your distinction between 'calculus limit' and real number is irrelevant. 1/3 is a number. It has the decimal representation 0.333. . . Labeling the latter as a 'calculus limit' in order to dismiss the legitimate use of the infinite decimal expansion is simply a semantic game without basis in logic.
First off, even in mathematics to say that 0.333... = 3 is nothing more than a semantic game. Give me an concrete example where this could even be used in a practical real live situation. It can't because it's meaningless to even speak of 0.333... existing in any real world sense.
So you all you are doing is arguing for a mathematics that is itself nothing more than semantic games.
Secondly, this question has absolutely nothing at all to do with the question of Cantor's diagonal proof. Even if 0.333... does equal 1/3 exactly that still doesn't help Cantor's problem.
Set theory is part and parcel of this whole question. The whole point is to prove the set of real numbers has a different cardinality than the set of natural numbers. Cardinality is a concept in SET THEORY. It is defined using bijective functions between sets.
But Cantor isn't making a bijection between sets. He's attempting to make a bijection between the set of natural numbers AND a numerical representation of decimals.
Not only this, but if he is going to claim this his "list" represents the set of decimals then we can see that his set is incomplete. Precisely because his list of decimals is necessarily incomplete because it's necessarily a square list. Therefore there should be no surprise at all that he arrives at the absurd conclusion that he is creating numbers that aren't on the list. Of course he would. That's exactly what his method much do.
So it's a false representation of a bijection of sets. His list is not the same as the actual set of decimal numbers.
Therefore if he's trying to make a bijection between sets his method fails miserably as I have been describing.
Cantor's argument is based on an assumed (for purposes of contradiction) bijective function between sets. The argument depends on the fact that real numbers have decimal expansion representations, and many (most!) numbers have infinite decimal representations. The bijective function can be considered even without considering how the real numbers might be represented.
I agree. I'm not renouncing the bijective function. A bijective correspondence can indeed be made between two sets. However, in this case that's not what Cantor is doing. In this case his representation of the set of decimal expansions is incomplete. Therefore it's not a valid bijection between sets.
Again, the main problem is your insistence without any logical basis in dismissing infinite decimal representations.
This is an incorrect statement of my position. I have never dismissed infinite decimal expansions. All I'm saying is the list that Cantor is construction is not a complete list of decimal expansions. Period.
So when he claims to be creating numbers that aren't on his list it's a meaningless claim. Just because he is creating numbers that aren't on his incomplete list does not mean that those numbers do not exist in the actual set of decimal expansions.
He's claiming to be creating numbers that aren't on the list. And that's his error.
OK. I will accept you have now clarified that your use of the word 'complete' corresponds to my second description. Using your now clarified definition, my list is not complete. However, the question is whether your notion of complete is relevant to Cantor's proof.
Finally some progress.
Cantor's entire proof requires that "his list
" contains all possible decimal expansions. Why? Because he's claiming to be producing a number that cannot be on the list. Therefore if his list does not contain all possible decimal expansions then his proof is meaningless. The fact that he can create numbers that aren't on 'his list' says nothing about the actual set of decimal expansions.
So his proof fails to show what he had hoped to be showing.
Apparently he didn't catch this, because he just assumed that lists of decimals could be listed in a square list.
Why does his list have to be square? Because he's using a diagonal line to create then numbers he's claiming are not on his list. But that method doesn't work for rectangular lists. Why? Because his diagonal cross-out-line simply can't descend down the list at a fast enough rate to cover all the possible decimal expansions. This is an innate property of the numerical representation of numbers that Cantor apparently never even gave a thought to. He was so concerned with making an abstract bijection between sets that he failed to realize that his diagonal line construction cannot be used to make this bijection.
'Squareness' simply has no definite meaning with infinite lists. Even if it did, a list not being 'square' is irrelevant to Cantor's argument since each number is represented by an infinite decimal expansion.
You seem to be too hung up on the geometric idea of a square. Perhaps it's that term that is confusing you. The point is that any complete list of numbers expressed using numerical symbols must necessarily have more rows than columns. Ignoring this fact in an attempt to imagine a geometrically square list does nothing to address the core issue.
The fact that Cantor can produce numerical representations of numbers that aren't on his list is meaningless, because his diagonal line method cannot cross off more than one digit per row. And that's simply not enough to perform the task he needs.
Your response is a nonsequitur. The sets in question have actual numbers as their elements. Cantor's argument involves a bijective function between sets of numbers. I have no idea what 'two different answers' you are accusing me of here. Two different representations of the same set still represent one set.
But Cantor's list of decimal representations cannot be valid representation of the set of all real numbers.
Let's not forget that he is claiming to be creating numbers that aren't on the list (i.e. aren't in the set real numbers. Talk about something that is nonsequitur.
You have several errors here. Cantor does not 'cross digits off.' He constructs a number by selecting digit N to be different than digit N in the Nth number in the assumed (for purpose of contradiction) bijective function. He says nothing about his process resulting in a 'square.' The 'squareness' of the finite representation of his infinite list is irrelevant and nothing more than a vague intuitive notion.
He does indeed 'cross digits off' why do you think they call this the diagonal line proof?
He constructs his number by crossing off digits and replacing them by his free choice of other numerals. And what other numerals is he choosing? He's choosing from numerals that make up the very notation that he is using. But he hasn't taken into account that those numerals require more rows than columns to fully describe these numbers.
He doesn't need to mention anything about "squareness". In fact, it's pretty clear that he ever even realized this folly. The squareness of his construction comes from the fact that he can only cross off one digit per row. So he's stuck with that whether he likes it or not.
And of course it follows that he'll create numbers that aren't on the list. That's naturally going to happen in any case because complete numerical lists of numbers necessarily require more rows than columns. So if you go down any complete list of numerals constructing a new number using a diagonal line that crosses off 1 digit per row you will ALWAYS produce a number that isn't on the list above where you are currently working.
In other words, this method of creating new arrangements of numerals is innate to the process he's invented. It has nothing at all to do with the actual properties of any sets. Yet he's claiming to have made a valid bijection between sets. But he hasn't done that at all. All he has managed to do is ignore the innate property of complete lists of numerals.
Your claim about the number he creates already being on the list is faulty reasoning based on your invalid assumption that what works for finite lists of numbers with finitely many digits must necessarily apply to what Cantor is doing with an infinite list of numbers with infinitely many digits.
Pretending that this process could be legitimately taken to infinity is a grave error.
I have already shown how and why Cantor's list get's further and further behind the real list of decimals with every digit he crosses off.
If anything he's getting infinitely behind the set of reals.
How in the world can you expect to take a process out to infinity when the process gets further and further behind with ever step you take?
In order to show that something works at infinity you need to show that it works for every step on the way toward infinity and there is nothing that would prevent it from continuing.
You can't just claim that a process that can't even be made to work at all will somehow magically correct itself at infinity. Where is there any justification for that?
I showed why this thing can't even get off the ground, but you want to take it to the edge of the universe and claim that by the time it gets there everything will magically be ok.
Where's the justification for that?
If you can't even get it off the ground you aren't going to make any journey to infinity.
This again is an error on your part as what is true for finite lists of numbers with finitely many digits need not be true for infinite lists.
It's hardly a proof of anything if the only thing you can claim about it is that it will magically repair itself at infinity if we simply ignore the fact that it doesn't even work at square one.
You really should go read the Mathematical Cranks book. It covers a lot of discussions like the one we are having. It documents how even a well-meaning amateur can go off the rails by continuing to insist that their thinking must be right while all the mathematicians must be wrong. Many of these examples exhibit the type of errors in thinking you are exhibiting here. Again, I hate to be unking, but that is the reality of this situation.
Sorry, but even the most prominent mathematicians fully realize that there are major problems with our mathematical formalism. They discuss these issues all the time.
In fact, mathematicians can't even agree on whether our mathematical formalism is invented or discovered.
Here's a trailer of a recent 3-part documentary on precisely this topic.
If you think that mathematics is fully understood, or even truly trustworthy in all of its claims, you really need to rethink that.
I've already answered many of the questions that Hanna Fry asks in this video. And it's also true that many mathematicians don't even agree on many of the things that the mathematical community has chosen to include in the formalism.
So if you think that mathematicians are all in agreement on these things, then you need to stop reading books about cranks and start reading books about real mathematicians. You'll quickly find that they have vast disagreements on many mathematical issues.
Mathematics if far some being carved in stone. And mathematics, being human, are indeed prone to making mistakes.
Our ideas about abstract concepts such as infinity are indeed quite abstract, and there is no reason to accept that these ideas are anything more than the imagination of men.
In fact, I'm not the only one who questions Cantor's ideas. There are actually quite a few mathematicians that don't take his ideas seriously. And rightfully so. Cantor treats infinity as though it is a finite quantity. How utterly absurd is that?