I'll bet that it's a big hit with the ladies on the bar scene, though. :DYou know the old joke. "Why can't you cross a mountain climber with a mosquito? Because you can't cross a scaler with a vector." That joke depends on conflating the engineering definitions of scalar, vector and cross (as in cross product) with the common English meaning of a climber -- a "scaler" -- and the medical meaning of vector -- a means of disease transmission, and the biological meaning of cross, as to cross-breed living things based on their genetic makeup.
But this is a JOKE, not something you can take seriously in a philosophical discussion. — fishfry
Countability as defined in mathematics simply has nothing at all to do with the everyday meaning of the ability to be counted ... If you counted, in the sense of saying out loud "one, two, three ..." the natural numbers, starting at the moment of the Big Bang, at the rate of a number per second; or ten numbers, or a trillion -- you would not finish before the heat death of the universe ... You can't count the natural numbers in the every day meaning of the word. — fishfry
given enough time, someone or something could count up to and beyond any arbitrarily specified value. — aletheist
Really? Cantor proved the reals constitute a continuum. Whatever they are, they are certainly not discrete. — tom
f you think to yourself, "The natural numbers, the integers, and the rational numbers are examples of foozlable sets," you will not confuse yourself or others by shifting the meaning of a technical term to its everyday meaning. — fishfry
Again, incorrect. You evidently have a rather idiosyncratic personal definition of "infinite." My dictionary provides several widely accepted definitions, and none of them state or imply that it means "not countable." Besides, as I keep noting, the concept of being "countably infinite" is well-established and well-understood within mathematics. — aletheist
Now you seem to be confusing "countable" with the idea of being finished counting. — aletheist
I have acknowledged this repeatedly - the natural numbers (and integers) are not actually countable, in the sense that someone or something could ever finish counting them. However, they are all countable in principle, in the sense that there are no natural numbers (or integers) that are uncountable; given enough time, someone or something could count up to and beyond any arbitrarily specified value. As I said before (with sincere gratitude), you have stated more accurately what I meant all along. — aletheist
Countable means capable of being counted. If it cannot be counted, as is the case with something infinite, or endless, it is not capable of being counted. Therefore the infinite is not countable. — Metaphysician Undercover
I think the issue here has been metaphysical — apokrisis
You do agree they're foozlable, right? I just want to make sure I'm understanding you. — fishfry
Yeah sure, that's the name you gave instead of the name "countable". But I'm not sure that I would agree with the assumption that there is a substantial difference between a foozlable infinity, and an unfoozlable infinity. We can call them countable and uncountable infinities if that's easier. — Metaphysician Undercover
The reals in their usual order are a continuum. They can be reordered to be discrete. Counterintuitive but set-theoretically true. — fishfry
The rationals in their usual order aren't discrete, — fishfry
These are murky philosophical waters. — fishfry
But if you insist that "countable" is to be used with its everyday meaning, then we should be careful not to confuse this with foozlability, which is a technical condition used by specialists in set theory. — fishfry
Once you conflated the technical meaning of countable with its every day meaning -- a logical fallacy -- the thread lurched off on a very unproductive tangent IMO. — fishfry
It is far from clear that "given enough time" you could count to any specified value. If time itself is part of the universe, then you will run out of time between the Big Bang and the heat death of the universe. — fishfry
You have just conflated counting up to some big finite number with counting ALL the natural numbers. — fishfry
I don't know if it's a "substantial" difference. It's certainly a difference. The rationals are foozlable and the reals aren't. Even in countable models of the real numbers, and yes such things exist, the reals are not foozlable. So yes it's a pretty important difference in math. — fishfry
If it is possible in principle to count up to any particular natural number (or integer), then it is possible in principle to count all of the natural numbers (and integers). — aletheist
I have done no such thing. I have noted, rather, that no matter how big a finite number you specify, it is possible in principle to count up to and beyond that number. In other words, you cannot identify a largest natural number (or integer) beyond which it is impossible in principle to count. If it is possible in principle to count up to any particular natural number (or integer), then it is possible in principle to count all of the natural numbers (and integers). — aletheist
This is a textbook case of the fallacy of composition. And, you've also forgotten one premise here, that any particular natural number has numbers higher than it. And, this is the premise which makes it impossible, in principle to count all of the natural numbers. — Metaphysician Undercover
Cantor proved the reals constitute a continuum. — tom
I have noted, rather, that no matter how big a finite number you specify, it is possible in principle to count up to and beyond that number. In other words, you cannot identify a largest natural number (or integer) beyond which it is impossible in principle to count. — aletheist
Countable means capable of being counted. If it cannot be counted, as is the case with something infinite, or endless, it is not capable of being counted. Therefore the infinite is not countable. — Metaphysician Undercover
But it is false to claim that the entire infinite set is countable in principle, what is countable is finite subsets. — Metaphysician Undercover
This is a textbook case of the fallacy of composition. — Metaphysician Undercover
He certainly did not think so. Could you please point me to the proof? Note, I acknowledge that the real numbers serve as a useful mathematical model of a continuum. — aletheist
If you think you can disorder the reals, then pleas indicate the number following this one, and suggest between which two numbers you might place it: — tom
I mention this because it's a counterexample to the intuition that a set can be "counted" if its members can be lined up so that there's one after another. — fishfry
No, you claimed the reals can be disordered — tom
Does mathematics actual model a continuum? I don't think so. — Rich
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