Wtf, Phylo- the question can be resolved rather simply, and that is the confusion of limits ,confused over confounding two kind of limits/infinities with one kind of mathematical conception of defining the problem. I know it sounds easy but it took a thousand years ,that even Galileo could not solve.
Well, would you consider it reasonable (if not a bit tautological) to say that:
For cases of sets P and K, when;
-
P = {1}
N = {1}
bijection = true -
P = {1,2}
N = {1,2}
bijection = true -
P = {1,2,3}
N = {1,2,3}
bijection = true -
P = {1,2,3…}
N = {1,2,3…}
by induction,
bijection = true
Thus by induction, because the fourth case is of infinite sets, bijection holds true for infinite sets.
??
I’m a little confused by this. I assume ‘K’ a typo, since you use N throughout, and that P and N aren’t intended to be the same set, but that the numbers in the set are meant to be placeholders for e.g. element 1, element 2, etc., so that P-element 1 is not necessarily the same as N-element 1. Is that right?
What is this intended to show? The induction seems to show that, if a bijection exists between two sets, and we add the same number of elements to each set, there will still be a bijective function between them. But I thought the intent was to show that, given that there is a bijective function between two sets, we can conclude that the two sets have the same cardinality. I don’t think this induction shows that. If that wasn’t the intent, can you clarify how this moves us forward?
I think a better tack is a proof by contradiction:
- Given: two sets P and N have a bijective function f(x) between them.
- Assume that P and N do not have the same cardinality.
- If they do not have the same cardinality, then there would be at least one element in the larger set that does not have a corresponding element in the smaller set.
- But then f(x) could not be a bijection.
- Since we know f(x) is a bijection, we can reject the assumption that P and N don’t have the same cardinality.
(\therefore) P and N have the same cardinality.
If that seems tautological, I agree. “Having the same cardinality” (\equiv) “Having the same number of elements” (\equiv) “Having exactly one element in one set for each element of the other set” (\equiv) “There is a bijection between the sets”. (EDIT: upon reflection, it may only be true for ordered sets that two sets of the same cardinality have a bijective function between them, in which case the statements aren’t equivalent)
A question for the room: Do you agree that it’s possible to articulate a number system in which (.999… = 1), and one in which (.999… \neq 1)? One of Phyllo’s links had a link to a paper that makes that case, seeming to argue from ambiguity, i.e. that .999… might be interpreted as being silent on the hyperreal places.
Let’s face it: All of this is tangled up [perhaps forever] in whatever actually is that elusive – illusive? – theory of everything.
And how that makes “nature” – reality, existence – what it is.
Or, in other words, not some other way instead.
Here everything would seem to be intertwined in everything else in a way that either subsumes “I” or does not.
Me, I am ever and always curious about the manner in which a particular point of view bears any “practical” consequences for those who think one way rather than another.
That way we can talk about whether any particular one of us is obligated to think in a particular way if they wish to be thought of as a rational human being.
So, when you argue that…
I can’t help but wonder if others might we thinking, “No, we don’t know that.”
Of course they may be thinking that. But are they presenting an argument?
The Planck scale presents a problem (IMO of course) for any such argument. But I’m open to hearing such an argument.
Note that JSS agrees that we are discussing a problem of math or logic and not one of physics. When JSS and I agree that must count for something 
Perhaps those who can solve this problem as pretaining to those below the Plank scale type of applications aren’t even human, if you have to get philosophical about it, this solution would solve the missing element, of disapplication of non physical c
onsiderations. Why not?
It has been a mere few hundred years that God has ’
died’ , and we have invented superior gods of steel ,
of machine.
How can a legitimacy of the age of the modern age
be verified?
How can the modern mathematician ever go below the threshold of the Plank threshold wherein it is
considerable only with assuming a hypothetical
scenario? Only by the presupposition, that either set has solution in terms of one and the other, where that solution has to be understood somewhere along
the line. Isn’t that the reason CERN was built at an
incredible cost, to find the physical manifestation to that assumption?
Trans-infinitessimals, as supported by Poincarre and others, is only a creation of Cantor’s imagination, demonstrated with the bisection of infinite sets.
But that does not deceive others to reduce the so called Liar’s Paradox from functional to earliest levels,
suggesting Wittgenstein’s critique for a satisfactory foundation using similar structural content within.
No.
Try again.
Stop being condescending, James, you haven’t earned it.
Stop being judgmental. YOU haven’t earned it.
There was nothing condescending in my post. Stop reading it in.
You misunderstood my notation. I corrected that. Try again.
There was no “adding twice” to anything.
The series shown depicts small very obviously equal sets growing in size then jumping to infinite length.
If you have two sets that are obviously equal in elements and size, P and N, bijection is true. By induction, the elements and size can be extrapolated infinitely without the trueness of bijection being affected.
From that series, the mind can easily accept that if it is true for the smaller sets getting larger, they could be infinite and still be equal and their bijection still be true (unless merely bickering online).
I asked for you to tell me why you believe that bijection works for infinite sets. You wouldn’t tell me, so I proposed something for you. Do you agree with it?
You’ve basically shown that you don’t understand high school mathematical induction.
Don’t tempt me.
To do what? Learn about mathematical induction?
… to expose the severity of your ignorance.
Dude what is the matter with you?
Induction is a way of proving that some proposition (P(n)) holds for all finite values of (n). So if you’ve proved (P(1)) and you’ve proved that (P(n)) implies (P(n+1)) then you can conclude that it’s true for all finite values of (n).
It says NOTHING about infinite values, nor are infinite sets even required to prove or use mathematical induction. It follows from the Peano axioms, which do not require the axiom of infinity.
You are lashing out because you just exposed your ignorance of material commonly taught to high school students. Wouldn’t you rather be interested in learning something rather than just flinging your poo?
I’m with wtf on this. That isn’t an inductive proof. For that, you show that, if something is true for n, it’s true for n+1, and then show some specific n for which it’s true (flashbacks of the Blue Eyes Problem…).
And I honestly don’t know that you can go from that, using finite n’s, to conclude that’s it true for infinite n’s.
But more importantly, to the extent it proves anything, it doesn’t prove anything interesting. If P and N are the same set, and you do the same thing to them, they remain the same set. And since a set always has a bijection to itself, it will always have a bijection to itself.
This seems different from what you’ve said before:
Are you saying that “bijection works” and “bijection proves equality of cardinality” are equivalent? It’s trivially true to show that bijection “works” for infinite sets because, again, a set always has a bijection to itself, namely the identity function f(x)=x.
Then to BOTH of you.
I am NOT talking about “mathematical induction”. Where in the hell do you think mathematicians got the notion of “induction” in the first place. In math, “mathematical induction” refers to a very specific use of induction. Fine.
WE ARE IN PHILOSOPHY wherein “induction” means to build an inference based upon similarity concerning ANY and EVERY subject.
Now get back to the series, as much as you hate to agree to anything, no matter how mind-dumbingly simple it is;
Do you now disagree that sets N and P are equal when infinite?
… and bbl
We are talking about math here, so using mathematical procedures seems justified.
And demands for rigor also seem justified. I’m not trying to be pedantic, I really don’t think I understand what you’re asking.
You’ve stipulated that P and N are same set, then you changed both of them in the same way. Yes, of course they’re equal.
That just begs the question: How do we go about demonstrating that the definitions that we give to the words used to explain our positions on this thread are in fact in sync with that which all rational men and women are obligated to share?
In other words, that there is a manner in which to determine definitively if in fact 1 = 0.999. And that my own analysis [and only my own analysis] reflects this.
All I’m doing – my “thing” here – is pondering the extent to which this may or may not be of some practical importance to us in our daily interactions with others.
Or, perhaps, 98.999% good enough.
Yes, that seems reasonable to me.
Again, I’m just pondering the distinction between the stuff that preoccupies me in the is/ought world, and the stuff that would seem to be squarely embedded in the either/or world. It would seem that mathematically 1 either does or does not equal 0.999. And then the philosopher would take that answer and probe the “meaning” of it. Say, ontologically. Or even teleologically.