That’s making a big assumption. What if Race Bannon used a nonstandard construction?
Nope. It’s not there. There is not even a tiny difference between the two examples.
I think you’re not worrying enough.
If Race was using a non-standard construction, he should have said so. Under the standard construction, Z is a set of equivalence classes on N[sup]2[/sup], Q is a set of equivalence classes on Z[sup]2[/sup], and R is a set of subsets of Q. Since 1[sub]Z[/sub] and 1[sub]R[/sub] are of different type, they’re not equal.
If you have a construction that creates the even integers separately from the rest of the integers, or one that creates the reals directly from the set axioms, then yes, there is no difference between the two examples. Otherwise, there is.
What else would the subscripts mean?
I understand what you’re saying, but those may (or may not) be standard–but they’re not common. Natural numbers are usually considered a subset of the integers, whereas under those definitions, they’re not.
There’s also no difference if one consideres even integers to be a subset of the integers, and the integers to be a subset of the reals–which is commonly done.
Natural numbers are usually considered a subset of the integers because they are isomorphic to a subset of the integers.
That, and they have the same names and numerals and are used interchangeably, except under those standard definitions.
RM Mentock and ultrafilter - I’m not sure I have a “side” on this, I’m just asking questions. I wouldn’t know what to call “standard”.
But let me take it this direction: I’ve heard it said that you can construct the reals from Dedikind cuts and also from Cauchy series (sequences?). I take that to mean that they’re alternate definitions, just two different ways of defining the same thing.
Now, after listening to ultrafilter, I am beginning to think that you’d claim that the “dedikind” number 2 is not the same as the “cauchy” number two, because one is a set, and the other is the limit of some sequence. I don’t know what to make of that.
My head is starting to hurt. If I think about this any more, I probably won’t be able to balance my checkbook.
You’re correct: the Cauchy sequence real 2 is not the same as the Dedekind cut real 2. Don’t think about it too much, or your head will hurt.
ultrafilter, I understand what you are saying. But when I think of the reals (or the rational or the integers), I think of any set that has the properties of the reals (or . . .). The set of Cauchy sequences and the set of Didekind cuts are both models for the real numbers.
These models for the reals have properties that we don’t assume that the real number have. For example, a Didekind cut is an ordered pair, we can talk about the set that is its first coordinate. Or if you consider a Cauchy sequence to be a real number, you can ask “What is the 15th term of the sequence pi?”
I guess that the way I look at it when you construct the real numbers, you aren’t really constructing the real number, but a model for the reals and any set that is isomorphic to the reals can be called the reals.
It just seems kind of silly to say that the set of integers is not a subset of the set of real number.
This whole thing reminds me of a paradox i once read about. How can anybody ever walk from one point to another? In order to reach their destination you have to go half the distance. Then you have to cross half of the remaining distance. then you have to cross half of that remaining distance… No matter how far you go, you always have half of the remaining distance to cover. But as anybody who has ever walked anywhere knows, eventually you do reach your destination. It is kind of the same thing with this problem. Yes, theoretically there will always be a number between 0.9999… and 1 (actually, there is an infinite number of numbers). but in order to have a number in between the two you have to stop the sequence somewhere, at which point you no longer have 0.999… If the sequence is extended out to infinite then you do eventually reach one, just like you eventually reach your destination.
That’s Zeno’s Paradox, which has already been mentioned about infinity - one times in this thread.
Isomorphisms are equivalence relations! They are equivalence relations defined on the one set of all algebraic structures under consideration (no matter what their underlying sets are). So you can have equivalence relations “between two” sets (groups, etc. ) if those very sets are elements of the same superset.
Therefore, it seems that, since our isomorphisms yield equivalence classes, we can consider 1 (real) and 1 (integer) to be equal in the same sense that 4 and 6 are “equal” mod 2.
Say what? An equivalence relation on S is a subset of S[sup]2[/sup] with the properties of reflexivity, symmetry, and transitivity. What the hell’s transitive about any isomorphism?
If R[sub]1[/sub], R[sub]2[/sub], and R[sub]3[/sub] are relations, Then we have:
Reflexive: R[sub]1[/sub] is isomorphic to R[sub]1[/sub]
Symmetry: If R[sub]1[/sub] is isomorphic to R[sub]2[/sub], then R[sub]2[/sub] is isomorphic to R[sub]1[/sub]
Transitive: If R[sub]1[/sub] is isomorphic to R[sub]2[/sub] and R[sub]2[/sub] is isomorphic to R[sub]3[/sub], then R[sub]1[/sub] is isomorphic to R[sub]3[/sub]
By “isomorphism,” I mean the property “being isomorphic” rather than the function which establishes the isomorphic correspondence. My profs used the word both ways.
Let S be a set of groups, for example. The subset of ordered pairs (A, B) of elements of S in which A is isomorphic to B forms an equivalence relation.
The transitive property holds. If A is isomorphic to B and B is isomorphic to C, then A is isomorphic to C. For the composition of two isomorphisms is itself an isomorphism, showing that A is isomorphic to C.
I am sorry if that have offended you, ultrafilter. I respect you very much. For over a year I have read and admired your posts. You always post something that comes from a knowledgeable and thoughtful mind. I wanted clarification on one point that I didn’t quite understand, and I hope you realise that I meant no ill.
–ragerdude
First, a nitpick. [nitpick]Under the Cauchy construction, real numbers are not sequences; rather they are equivalence classes of sequences. So π does not have a 15th term, though each of its elements does.[/nitpick]
Aside from that nitpick, DrMatrix has nicely stated my view on this problem of whether 2[sub]Z[/sub] is the same thing as 2[sub]R[/sub]. Mathematicians are accustomed to constructing models in Zermelo-Fraenkel set theory of mathematical objects, but we can (though perhaps rarely) find ourselves in ontological trouble if we identify our models with the objects we are studying.
For example, consider the ordered pair (a, b). One can model this ordered pair in ZF as the set {a, {a, b}} or as the set {{a, b}, b}. Saying that there are two different number 2’s (e.g. 2[sub]Z[/sub] and 2[sub]R[/sub]) is like saying that there are two different ordered pairs (a, b), one for each of the two conventions above. But no mathematician thinks that, given a fixed a and b, there is more than one ordered pair (a, b). We might be aware that there are these two conventions of representing this ordered pair in ZF, but, in practice, mathematicians are rarely called upon to choose either convention. For us, an ordered pair isn’t a set at all. We think of an ordered pair as a different sort of object from a set. It is nice that there is a theory like ZF that it can model ordered pairs, but this fact doesn’t tell us what ordered pairs are; it just tells us that the concept of ordered pairs is not a contradictory one (at least not if ZF is consistent).
Likewise, it is good thing that real numbers have been constructed by Dedekind cuts, Cauchy sequences, and other such ZF constructs. But the primary value is in knowing that the concept of an ordered field with the least upper bound property is consistent. These constructions shouldn’t be thought of as telling us what the real numbers are.
So, just what is R, if it is not one of these constructs? One’s answer to this (if one cares to think about it at all) depends on the ontological theory of mathematics to which one subscribes. Possible answers include the following.[ul]
(1) R is simply some particular abstract object, which we apprehend directly, and which shares some important structural properties with certain objects that may constructed in ZF. The number 2 is some sort of piece or element of this object R (but here “element of” is not necessarily being used in the sense of ZF).
(2) R is the class of all abstract objects that share the relevant structural objects, including the constructions from Dedekind cuts and Cauchy sequences. The number 2 is the class of objects that serve the role of “2” in one of these constructions.
(3) R does not exist as an object at all. There are just certain propositions of the form “R has property blah” that may be derived within certain important formal systems.[/ul]
Sorry, you caught me in a bad mood. I didn’t mean to be harsh. Please, accept my apologies and thanks for your kind words.
I don’t have time to get involved in this discussion right now, but you should notice that in your example, we’re looking at an equivalence relation on the set of groups, not between any two groups. In that situation, you can’t say that any element of a group A is equivalent to some element of a group B, even if A is isomorphic to B. Does this make sense?
As an aside, Tyrell McAllister has a very good point. It’s probably more proper to say that 1[sub]Z[/sub] is not necessarily identical to 1[sub]R[/sub], just because you can have different models for Z and R.
I should say that item (2) in the list in my last post should readul R is the class of all abstract objects that share the relevant structural properties, amongst which are the constructions from Dedekind cuts and Cauchy sequences. The number 2 is the class of objects that serve the role of “2” in one of these constructions.[/ul]
Ever one to throw in my two cents on touchy issues I feel I’ve finally grasped (I’m just waiting for a good Monty Haul Problem discussion to break out at the next party I go to)…
It seems to me that there are 10 kinds of people here, those that understand binary and those that don’t.
Just kidding (stole it from a shirt, which likely stole it from someone else), but it does approach the heart of the matter. I think there are just a few basic camps of folks here, and it hinges on what they believe the following value is:
The limit as x approaches 1 of x + 1
Anyone in the first camp with a firm grasp of limits will tell you the value of the above statement is EXACTLY equal to 2.
Those in the second camp that are almost there and just need to have their hands slapped with a slide rule a few more times will tell you that the value of the above statement is ALMOST, NEARLY, REALLY REALLY CLOSE, SO CLOSE THERE’S NOTHING CLOSER, to 2 but NOT EXACTLY 2. They’re wrong, but bless their hearts for staying at least half awake during match class. Some will eventually come to realize this, others will defend to their dying breath that a limit never reaches its destination. I think this thread is as close as possible to rescuing those who can eventually understand that, but I’m posting just in case there are any left.
To try and set straight another misconception that has popped up:
“You can’t do math with infinities running around”.
It is true that our four basic mathamatical operations (+, -, X, and /) do fail with a few VALUES, namely division failing with a 0 value for a denominator, and pretty much all four of them failing if an infinate VALUE is used (not a FINITE value with an INFINATE representation). Other people may argue (and perhaps correctly) that these two are the same issue, or maybe try to raise a few more, but in general I find these two simple principles to work with and they can be reduced or expanded to any other system.
For example, if I write:
x + 1 - x = 1
The equation will hold for x = 0, x = 1, x = PI (even a theoretical infinite decimal representation of it), x = 1/3, x = 0.3…, and even x = 0.9…
The equation is not valid if I claim x = 1/0 or x is infinately large. So some “proofs” fail, like:
x = 1 + 2 + 4 + 8 + …
x = 1 + (2 + 4 + 8 + …)
2x = (2 + 4 + 8 + …)
2x = x - 1
x = -1
(getting a negative value from adding all positive integers is just nutty)
because the VALUE of one of its terms (namely, x) is infinate and thus the four basic operations aren’t valid.
However, the proof for 0.9… = 1:
x = 0.9…
(multiply both sides by 10)
10x = 9.9…
(subtract the first equation from the second)
9x = 9
(divide both sides by 9)
x = 1
IS valid, because its terms (x, 0.9…, 10x, 9.9…, 9x, 9, x) are ALL finite values and there are no divisions by zero.
For those that are having difficulty grasping that an event which never occurs at any finite step can occur after an infinate number of steps, I proffer the following (perhaps overused) example:
1 + 1 + 1 + 1 … (adding an infinate number of 1’s) = infinity
not “really really close but not quite infinity”, it really does equal infinity. I feel a little uneasy about saying “equal infinity” to everyone, so let me say:
x = 1 + 1 + 1 + 1 + 1 + …
The value of x is infinately large
I hope I’ve got everyone, or very nearly everyone behind me in using those two statements together. Now take cases step by step:
1 + 1 = 2 (not infinately large)
1 + 1 + 1 = 3 (also not infinately large)
1 + 1 + (… 1000 1’s) = 1000 (still quite finite)
At every step in the equation, the value is not infinate, the same way that at evey step of 0.9, 0.99, 0.999 the value is not 1.
If you can accept that an infinate number of 1’s added together has infinate value even though it never happens after a finite number of steps, perhaps you can accept that 0.9… is has the same value as 1.
Just in case someone reads this thread in the future–Cecil’s response to the OP: An infinite question: Why doesn’t .999~ = 1?