01-02-2006, 06:57 AM
Will... to... resist... butting... in... failing...
A set is an unordered collection of elements. To say that a set "begins" with something is contrary to the very notion of a set. To say that a set even has direction is also so much nonsense. Consider: { (x,y) | x^2 + y^2 = 1 }.
Countability and cardinality are topics of introductory set theory. The axiom of choice is somewhat more advanced.
A set is unordered. A listing of them may have some "order" to it, but that's your doing. A relation may be able to help you put the elements "in order" somehow. For example, the set { x | 0 <= x <= 1 } with the relation < (or >) can tell you, for any two distinct elements, whether one is larger/smaller.
Integers can be negative. So, no.
I think that your attempt to link causality and listings of elements in a set contained more words than necessary. It often seemed that you were struggling to find the words, so you used a whole lot of them. The concept seems sufficiently simple that I think a truly monstrous simplification exists (e.g. "I believe that effect has a cause, in much the same way that any listing of a set has a first element."), so if you could make a post like that, it'd be neat to read it.
-Lemmy
wakim,Dec 31 2005, 11:42 AM Wrote:A glance at the example reveals that {. . ., -2, -1, 0, 1, 2, 3, . . .} seems to begin with negative infinity and end in infinity; yet a set infinite in both directions has no beginning and no end (if we both share this meaning of âinfinityâ), no first or last.
A set is an unordered collection of elements. To say that a set "begins" with something is contrary to the very notion of a set. To say that a set even has direction is also so much nonsense. Consider: { (x,y) | x^2 + y^2 = 1 }.
Quote:And, obiter dicta, isn't âaxiom of choiceâ and the phrase âcountably infiniteâ indulgences in advanced set theory?
Countability and cardinality are topics of introductory set theory. The axiom of choice is somewhat more advanced.
Quote:If one supposed an ordered set (analogous to possessing cause - an ordering principle), shouldn't one then expect a definitive first element?
A set is unordered. A listing of them may have some "order" to it, but that's your doing. A relation may be able to help you put the elements "in order" somehow. For example, the set { x | 0 <= x <= 1 } with the relation < (or >) can tell you, for any two distinct elements, whether one is larger/smaller.
Quote:The question of: âIf a set contains an infinite number of integers, mustn't in contain every quantity of integers smaller than the infinite? In other words, if a set of 100 items were examined, mustn't it also be found to contain 99 items, and 98 items, and so on? If a set of 100 items did not contain a first item, how could it contain a second?â is the original example, given explicitly under the premise that âeverything... has a cause,â and further prefaced by this question of mine:
Integers can be negative. So, no.
Quote:If one accepts uncaused effects, then isnât it beyond any field of study to find the cause of those effects?
I think that your attempt to link causality and listings of elements in a set contained more words than necessary. It often seemed that you were struggling to find the words, so you used a whole lot of them. The concept seems sufficiently simple that I think a truly monstrous simplification exists (e.g. "I believe that effect has a cause, in much the same way that any listing of a set has a first element."), so if you could make a post like that, it'd be neat to read it.
-Lemmy