Guess what?

[Tunnelcat]

(can't speak for women, because I'm not one )--that's liberty.

"Cough"....."Sniiiiiiirk" But it might give you a whole new insight if your were to suddenly become one tomorrow. I'd be a learning experience. Your world outlook on power, dominance, oppression, liberty and freedom, even what you think God intends for us, would be a whole hell of lot different than what it is now. Not necessarily better or worse, just different.

[Ferno]

power? dominance? opression?

Those things for women don't exist in the states nearly as much as feminists think. Now go to a country like.. Somalia or NK where feminism is needed and you'll wake up.

[Tunnelcat]

Or any Muslim dominated country out there on Earth.

[Foil]

...I'm holding off.. until Foil replies to my earlier question, though.

This one?

... could an algorithm or equation of some sort be formulated that would describe, mathematically, a multiverse (set a) that contains all other universes including a paradoxical multiverse (set n) that contains every universe that shouldn't or couldn't technically exist within multiverse a? I feel like maybe I'm asking the same thing using different words, but I know there are numbers that cannot be expressed in our universe normally, but can be expressed using algorithms (Graham's number comes to mind). So I was wondering if the same could be done to mathematically describe a paradoxical multiverse.

Let's say you can define your self-contradictory / paradoxical set as above. The problem is, once you've done so, you can't do any rigorous math with it. If you try to use it, you're no longer working in set theory; you're off in some paradoxical non-mathematical arena (which I would know nothing about).

P.S. Graham's number doesn't work as an analog here, because it's not paradoxical or self-contradictory. It's just simply too huge to reasonably express with our usual methods of expressing large numbers.

[Ferno]

Or any Muslim dominated country out there on Earth.

Exactly. Feminists really need to take the fight to them.

[MD-1118]

...I'm holding off.. until Foil replies to my earlier question, though.

This one?

... could an algorithm or equation of some sort be formulated that would describe, mathematically, a multiverse (set a) that contains all other universes including a paradoxical multiverse (set n) that contains every universe that shouldn't or couldn't technically exist within multiverse a? I feel like maybe I'm asking the same thing using different words, but I know there are numbers that cannot be expressed in our universe normally, but can be expressed using algorithms (Graham's number comes to mind). So I was wondering if the same could be done to mathematically describe a paradoxical multiverse.

Let's say you can define your self-contradictory / paradoxical set as above. The problem is, once you've done so, you can't do any rigorous math with it. If you try to use it, you're no longer working in set theory; you're off in some paradoxical non-mathematical arena (which I would know nothing about).

P.S. Graham's number doesn't work as an analog here, because it's not paradoxical or self-contradictory. It's just simply too huge to reasonably express with our usual methods of expressing large numbers.

Yes, that's the one. Thank you for clearing that up for me. I have one more question relating to the concept of infinity. Am I correct in visualising the difference in "sizes of infinity" (worded improperly, I'm sure, but hopefully it gets the idea across well enough) by referencing the difference between a line and a ray in geometry? One being infinitely long, with no end in either direction, the other also being infinitely long, but with a definite starting point.

On the topic of the multiverse, and modal realism... there was an article on TOW concerning Max Tegmark's Ultimate Ensemble, but I can't find it now. His 'Ultimate Ensemble', or level IV multi-/omniverse, contains all other possible levels and instances of universe by overgeneralising. The TOW article on modal realism has this to say:

Possible worlds exist – they are just as real as our world;
Possible worlds are the same sort of things as our world – they differ in content, not in kind;
Possible worlds cannot be reduced to something more basic – they are irreducible entities in their own right.
Actuality is indexical. When we distinguish our world from other possible worlds by claiming that it alone is actual, we mean only that it is our world.
Possible worlds are unified by the spatiotemporal interrelations of their parts; every world is spatiotemporally isolated from every other world.
Possible worlds are causally isolated from each other.

And:

Lewis backs modal realism for a variety of reasons. First, there doesn't seem to be a reason not to. Many abstract mathematical entities are held to exist simply because they are useful. For example, sets are useful, abstract mathematical constructs that were only conceived in the 19th century. Sets are now considered to be objects in their own right, and while this is a philosophically unintuitive idea, its usefulness in understanding the workings of mathematics makes belief in it worthwhile. The same should go for possible worlds. Since these constructs have helped us make sense of key philosophical concepts in epistemology, metaphysics, philosophy of mind, etc., their existence should be uncritically accepted on pragmatic grounds.

I also had a thought earlier that, unfortunately, is most likely lost to me forever, but fragments remain. Those fragments could be summed up thusly: "Alphabetical language and numerical language are two different, but equally valid, forms of communicating ideas." and "Numbers and letters are both symbols used to transfer meaning and information." There are a few other remnants of this thought I retain, but I do not know if I should entertain them. The reason I am not certain I should or should not entertain them is because I am not certain if the societal and scientific reasons for discarding such thoughts are, in fact, sound. The fact (heh) that science in its modern form is built on axioms that are, in and of themselves, assumptions does not sit well with me given how infallible the construct of science is considered by modern society. This, too, relates to the thought I had.

I know, I know, "nothing new under the sun" and all that jazz, but it seemed like a very interesting idea at the time, with far-reaching implications. Perhaps I should give my brain a rest until I fully recover from this head cold, I may not be thinking straight.

Then again, when am I?

[Foil]

I have one more question relating to the concept of infinity. Am I correct in visualising the difference in "sizes of infinity" (worded improperly, I'm sure, but hopefully it gets the idea across well enough) by referencing the difference between a line and a ray in geometry? One being infinitely long, with no end in either direction, the other also being infinitely long, but with a definite starting point.

No. In fact, there's a 1-to-1 mapping from the points on a ray to the points on a line, so they'd both be at the same cardinality.

[MD-1118]

I have one more question relating to the concept of infinity. Am I correct in visualising the difference in "sizes of infinity" (worded improperly, I'm sure, but hopefully it gets the idea across well enough) by referencing the difference between a line and a ray in geometry? One being infinitely long, with no end in either direction, the other also being infinitely long, but with a definite starting point.

No. In fact, there's a 1-to-1 mapping from the points on a ray to the points on a line, so they'd both be at the same cardinality.

Yes, but is it an injective non-surjective function, a non-injective surjective function, or an injective surjective function?

If there is an exact 1-to-1 correspondence (both '1-to-1' and 'onto') between L (line) and r (ray), then why make the distinction? It seems counter-intuitive.

[Foil]

If there is an exact 1-to-1 correspondence (both '1-to-1' and 'onto') between L (line) and r (ray), then why make the distinction? It seems counter-intuitive.

Rays and lines are different, simply by definition.

Also, you were the one who made the distinction by (incorrectly) suggesting there was a difference in the cardinality of their point-sets.

----------

In any case, I'm still not clear where you're going with these these ill-formed ideas (sets of paradoxical sets, and visualizing levels-of-infinity with rays and lines)... can you clear this up?

[MD-1118]

If there is an exact 1-to-1 correspondence (both '1-to-1' and 'onto') between L (line) and r (ray), then why make the distinction? It seems counter-intuitive.

Rays and lines are different, simply by definition.

Also, you were the one who made the distinction by (incorrectly) suggesting there was a difference in the cardinality of their point-sets.

----------

In any case, I'm still not clear where you're going with these these ill-formed ideas (sets of paradoxical sets, and visualizing levels-of-infinity with rays and lines)... can you clear this up?

I'm a visual thinker. This is what I see in my head when I think of L and r:

L: <---o--->
r: o--->

Visually, they are not the same thing, even if you discount their definitions. Visually, r has a definitive, concrete point of origin, whereas L does not. How, then, can they have the same cardinality? Again, is it an injective non-surjective function, a non-injective surjective function, or an injective surjective function? I don't understand the argument of 'different by definition'.

I'm sorry for being so obtuse. I am merely trying to ascertain whether I am looking at these things correctly, and according to your replies, I am not. In case you couldn't tell, math is not my strongest topic.

[Foil]

L: <---o--->
r: o--->

Yes, that's essentially what they are.

How, then, can they have the same cardinality?

They have the same cardinality (i.e. the same "level of infinity") because there's a bijection (i.e. "1-1 correspondence", "injective surjective mapping") from the points on one to the points on the other.

To see this, here's an analagous bijection: Consider the set of all integers x as the "line" and the set of all non-negative integers y as the "ray" starting at 0. There is a bijection from one to the other, so they have the same cardinality.

Now, you might think: "Wait, there's more points on the line!" Not true, when we're talking about the cardinality of infinite sets.

Similar exercises:

Which set has the larger cardinality? The set of all positive integers (1, 2, 3...), or the set of all even positive integers (2, 4, 6...)?

Which set has the larger cardinality? The set of all positive integers (1, 2, 3...), or the set of all positive integers except the first million (1000001, 1000002, 1000003...)?

[Top Gun]

To sum up, number theory is weird.

[Foil]

I highly suggest taking some good courses in this stuff.

[Top Gun]

I can't say that the more abstract side of mathematics, i.e. set theory and the like, has ever held much interest to me personally. One of the reasons I like calculus so much is that it's an incredibly-practical branch of mathematics, and is used as the basis for pretty much every real-world phenomenon we encounter daily.

[MD-1118]

L: <---o--->
r: o--->

Yes, that's essentially what they are.

How, then, can they have the same cardinality?

They have the same cardinality (i.e. the same "level of infinity") because there's a bijection (i.e. "1-1 correspondence", "injective surjective mapping") from the points on one to the points on the other.

To see this, here's an analagous bijection: Consider the set of all real numbers x as the "line" and the set of all non-negative real numbers as the "ray" starting at 0. The map f(x) = 2x is a bijection from one to the other, so they have the same cardinality.

Now, you might think: "Wait, there's more points on the line!" Not true, when we're talking about the cardinality of infinite sets.

Similar exercises:

Which set has the larger cardinality? The set of all positive integers (1, 2, 3...), or the set of all even positive integers (2, 4, 6...)?

Which set has the larger cardinality? The set of all positive integers (1, 2, 3...), or the set of all positive integers except the first million (1000001, 1000002, 1000003...)?

I was following you up until the bolded portion (emphasis mine). Why is this the case?

[Foil]

Here's a decent explanation, but really you just have to think through what cardinality means, and what a bijection does.

For example, in the first exercise, you can make a very simple bijective map from the (1, 2, 3...) set to the (2, 4, 6...) set by just doubling. Go through it, you'll find that the mapping is indeed bijective (everything in each set corresponds to exactly one thing in the other, both ways). If you start trying to think about which set is "bigger" or which one "runs out" first, you're not thinking about it correctly.

Same thing for the second exercise, just add a million to get a mapping from (1, 2, 3...) to (1000001, 1000002, 1000003...) which is 1-1 both ways. Remember that these are infinite sets, so if you starting thinking about smaller chunks of the sets, you're not looking at it correctly.

P.S. I fixed my earlier statement above, I was mixing up two different examples.

[MD-1118]

Here's a decent explanation, but really you just have to think through what cardinality means, and what a bijection does.

For example, in the first exercise, you can make a very simple bijective map from the (1, 2, 3...) set to the (2, 4, 6...) set by just doubling. Go through it, you'll find that the mapping is indeed bijective (everything in each set corresponds to exactly one thing in the other, both ways). If you start trying to think about which set is "bigger" or which one "runs out" first, you're not thinking about it correctly.

Same thing for the second exercise, just add a million to get a mapping from (1, 2, 3...) to (1000001, 1000002, 1000003...) which is 1-1 both ways. Remember that these are infinite sets, so if you starting thinking about smaller chunks of the sets, you're not looking at it correctly.

P.S. I fixed my earlier statement above, I was mixing up two different examples.

Okay, I think I'm beginning to understand. Basically if one assumes all elements of the (2,4,6...) set to have the same 'traits' (i.e. they do not have any distinguishing characteristics aside from being individual elements of a particular set), and one also assumes the same for all elements of the (1,2,3...) set, then they have the same amount of elements, period, because in the (2,4,6...) set, there are no odd numbers - they simply do not exist to be 'misplaced' or 'unaccounted for'. In other words, they have the same cardinality, but not the same order type. Is that correct?

[Foil]

You're over-thinking it. They have the same cardinality, simply because you have a bijection (a 1-to-1 map from one to the other, and back) which applies for the entire set.

Another way to illustrate it, in the (1,2,3...) / (2,4,6...) example:

Code: Select all

1 2 3 4 ...
^ ^ ^ ^
| | | |
v v v v
2 4 6 8 ...

Don't get confused by the fact the numbers are different in the latter set. As long as you have that bijective map between two infinite sets, they have the same cardinality ("size of infinity").

[MD-1118]

Basically if one assumes all elements of the (2,4,6...) set to have the same 'traits' (i.e. they do not have any distinguishing characteristics aside from being individual elements of a particular set), and one also assumes the same for all elements of the (1,2,3...) set, then they have the same amount of elements, period, because in the (2,4,6...) set, there are no odd numbers - they simply do not exist to be 'misplaced' or 'unaccounted for'. In other words, they have the same cardinality, but not the same order type. Is that correct?

Corrected. I misread the article I was referencing.

You're over-thinking it. They have the same cardinality, simply because you have a bijection (a 1-to-1 map from one to the other, and back) which applies for the entire set.

Another way to illustrate it, in the (1,2,3...) / (2,4,6...) example:

Code: Select all

1 2 3 4 ...
^ ^ ^ ^
| | | |
v v v v
2 4 6 8 ...

Don't get confused by the fact the numbers are different in the latter set. As long as you have that bijective map between two infinite sets, they have the same cardinality ("size of infinity").

The way I visualised this was as having two sets of apples, both infinite. Set N's apples are labelled (1,2,3...), whereas set 2N's apples are labelled (2,4,6...). Still the same (infinite) amount of apples, they're just labelled differently. Basically the point to looking at it this way is to remind myself, in cases such as this, that numbers themselves are treated as physical entities, 'set occupants' if you will, rather than abstract intellectual constructs or ideas (which is where the problem "but there are more numbers in between the even ones!" comes from - not realising this, because there aren't 'more numbers in between'. If you 'include' the 'missing numbers' from set 2N, for example, then you aren't considering 2N - you are actually considering N).

I highly suggest taking some good courses in this stuff.

I fully intend to... you've got my curiosity piqued. Thanks!

You're over-thinking it.

Story of my life... @@

[Foil]

The way I visualised this was as having two sets of apples, both infinite. Set N's apples are labelled (1,2,3...), whereas set 2N's apples are labelled (2,4,6...). Still the same (infinite) amount of apples, they're just labelled differently.

Nice. That's a good way to visualize it.

Back to your ray/line thing, it's the same kind of thing, though the bijection between them is more complicated.

[MD-1118]

The way I visualised this was as having two sets of apples, both infinite. Set N's apples are labelled (1,2,3...), whereas set 2N's apples are labelled (2,4,6...). Still the same (infinite) amount of apples, they're just labelled differently.

Nice. That's a good way to visualize it.

Back to your ray/line thing, it's the same kind of thing, though the bijection between them is more complicated.

Thanks. I feel like a small child who has finally learned how to ride a bicycle. Not well, mind you, just... without falling over.

Now for something more complicated. Why can't set a (the set of all sets) contain itself? Or can it?

[Foil]

Why can't set a (the set of all sets) contain itself? Or can it?

It's because that definition leads to a contradiction, at least under the normal set-theory axioms. (For example, it would have to include a set we already know is paradoxical: the classic "set of all sets which do not contain themselves".)

Best explanations I've seen.

That said, apparently there are alternate axioms which allow the universal set; I'm not familiar with the details, but it might be worth looking into, if you're curious.

[MD-1118]

Why can't set a (the set of all sets) contain itself? Or can it?

It's because that definition leads to a contradiction, at least under the normal set-theory axioms. (For example, it would have to include a set we already know is paradoxical: the classic "set of all sets which do not contain themselves".)

Best explanations I've seen.

That said, apparently there are alternate axioms which allow the universal set; I'm not familiar with the details, but it might be worth looking into, if you're curious.

I definitely am curious. Also, I clicked a link on the page you linked, and found the following:

The Barber Analogy


Let me tell you a story...

Once upon a time there was a town with strict laws on shaving. Everyone was required by law to shave daily. Those who didn't feel like shaving themselves went to the town barbershop. The barber who owned the shop had been legally appointed “the man who shaves all and only those who don’t shave themselves”. This was all well and good, until the barber was arrested for being unshaven. You might think the barber could have just shaven himself, but the law said that the barber could only shave those who don’t shave themselves. And if he didn't shave himself, he was supposed to be shaved by the barber. But he is the barber, and once again, the barber was not supposed to shave men who shave themselves! When the judge realized that it was impossible for the barber to follow the law, he ruled that the law be revised to resolve the contradiction and that the barber be released from jail.

The set of all sets which do not contain themselves is analogous to the barber who only shaves men who do not shave themselves. If the barber shaves himself, he is no longer a person who is allowed to be shaved by the barber; if the set of all sets which do not contain themselves contains itself, it is no longer allowed to contain itself. If the barber does not shave himself, he should be shaved by the barber; if the set does not contain itself, it should be contained in itself. This is Russell’s Antimony. Just as the law is impossible for the barber to follow, the set of all sets which do not contain themselves is a logical impossibility. See the image below for a visual presentation of the contradiction.

This seems to imply the fault lies with the law not being self-consistent because it does not account for the barber's case (this does not mean the barber must be jailed, but rather the law must be changed); this in turn would mean that the ZF set-theory axioms are incomplete (the universal "set of all sets" is not nonexistent, but rather ZF does not account for it).

[Foil]

So you're saying that the Zermelo-Fraenkel axioms for set theory aren't as good/valid as others, because they don't allow a universal set?

Okay, fine, but now you have to explain why it's somehow critically necessary to allow for the universal set. (Why do you need it? Is it related to your original topic?)

[MD-1118]

So you're saying that the Zermelo-Fraenkel axioms for set theory aren't as good/valid as others, because they don't allow a universal set?

Okay, fine, but now you have to explain why it's somehow critically necessary to allow for the universal set. (Why do you need it? Is it related to your original topic?)

They are good/valid in their own right, when applied to suitable situations, much like how special relativity and general relativity don't match up, but have their own applications under different frames of reference. And yes, I do need the concept of a universal set regarding the original topic. That's where Tegmark's Ultimate Ensemble, and Lewis' reasoning for modal realism, come into play.