Replies to M. Pka and M. HallsofIvy
M.Pka
OK, I have Henle’s book on order but it probably won’t arrive until late this week or early next week. I look forward to reading his descriptions and explanations. I withdraw my question above rather than try and explain further where I was trying to go with it, perhaps it will be unnecessary.
I recognize that the axiom of infinity provides the definition of an “inductive set” in which each element has a successor and a predecessor except zero which has only a successor. This I understand encodes Peano’s integers and is the first definition of an infinite set. This knowledge is of course only a recitation from reading that I have done.
Two other interesting things I have read regarding integers are first, that mathematics is largely an extension of and therefore is reducible to the invention of counting. (I imagine NASA keeping track of a space flight by employing ten thousand people who, in symphonic concert, count on their fingers and toes.) OK, OK, but I do see, in a hazy way, that counting provides the proto type inductive set, and goes on to form the basis of defining the integers, rational numbers, and, still more hazy, the real numbers … after that the moon.
Another interesting fact, so I have read, is that certain “primitive” tribes have only a word for one, two, and three, after that the quantity is “a few”, or “many”. Perhaps among small bands of people those quantities are descriptive enough for pursuing game or calling up the guard. … just interesting while adding a perspective to the enterprise of mathematics as a whole.
Anyway, Peano’s axioms for defining integers do make sense to me, and, as important I think, the idea he and others of his era had, that there was something to gain by abstracting the essence of what counting is and thereby look beyond integers for other sets of things that might be inductive in nature.
Hmmm, I am trying to think of whether induction applies to objects outside of mathematics. I know that inductive reasoning in science (generalizing from particular cases) is NOT the same thing … something to ponder while on the treadmill.
I did read that Wikipedia article on transitive sets. I think I understand that the essence of a transitive set is that if the elements of an overall set are sets, then the elements of those subsets, most significantly the “proper subsets”, are also elements of the overall set, For example, in set theory, the overall set of natural numbers N, is the set of integers each of which is defined as a set of integers, makes N a transitive set, e.x. 3 is defined as the set of the integers {0,1,2}, 3 as well as its constituents, 1,2, and 3 are all integers.
I was thinking that currency, thousand dollar bills, hundreds dollar bills, ten dollar bills might also be a transitive set … but that is just the integer case again in decimal form, counting by tens, (I think)….
Hmmm, is that the point, that there is really only one inductive set, only one transitive set, one set of rules per set type, that one investigates a particular set, firstly, by trying to map it to an existing set prototype, or else, if you really know what you are doing, by defining a new prototype. If so, this revelation for me is probably the sort of knowledge that everyone else has seen right off while watching me bump into walls. Ah well.
I am going beyond myself here, I think I will wait for Henle’s book before asking any serious questions about set theory.
M. Hallsof Ivy
Thanks, you wrote quite a bit. The question I asked question was a bump in the road. I was trying to decide whether an integer is represented by an ordered set, or an unordered set.
Take, 3, for example … 3 = {{0}, {1}, {2}}, an unordered set, but can {{0}, {1}, {2}} be represented as an ordered pair?
Since {{0}, {1}, {2}} = 2 U {2}, I was trying to take the next step of saying 2 U {2} = { 2, {2} }
Would that be a correct statement transformation, a name of a set, i.e. “a”, paired with a set, i.e. {2}. If so, would { 2, {2} } be an ordered pair. That is why I was asking what “{2}” is.
If {2} is a proper subset of 2, or vice versa, then 3 = {{0}, {1}, {2} } = 2 U {2} = { 2, {2} } is both an ordered pair and an unordered pair. QED Set theory crumbles to the ground (heh, heh), or, I am screwed.
So, trying to noodle my way out of the dilemma, 2 = {1,0} and {2} = {{1,0}} … is the latter a subset of the former, or vice versa.
In the first case only {1} and {0} are proper subsets of 2 and neither are equal to {{1,0}}. But, on the other hand, still cogitating, you could say that 2 IS a subset of {2}. Hmmm, maybe set theory and hence civilization are saved because 2 is not a “proper” subset of {2}, hence {2, {2}} is not on ordered pair and therefore 3 is an unordered set, period.
Well, once again I ask for confirmation or correction ... integers, in set theory, are defined as unordered sets, NOT ordered sets, perhaps along the line that I have reasoned
This is important because originally I had thought the opposite and hence thought ordered pairs vis-avis set theory were formulated to define the set of integers and not to reverse engineer a definition for an ordered pair vis-à-vis its characteristic property. My reasoning was that the nested predecessors of an integer constituted a singleton and as such were a proper subset of their successor element and hence an integer would be defined as an ordered pair.
{ a, {a,b} }
a = {singleton of nested predecessors},
b = { successor element}
a is a subset of b because all the elements in "a" are nested inside of b.
Anyway, perhaps I need to wait for Henle’s book and start from the beginning, perhaps this flapping around in just muddying the water.
