An uncountable countable set
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From: MoeBlee on 19 Sep 2006 13:45 Tony Orlow wrote: > Given N is the standard naturals: > > Actually infinite(s) = E seS A neN index(s)>n > Potentially infinite = A seS index(s)eN ^ ~E neN index(s)<>n First, that doesn't even make sense as a definition, since you have a free variable 'S' (upper case 'S') that is in the definiens but not in the definiendum. Next, what is the logistic system? What are the non-logical primitives? What are the non-logical axioms? What is the definition of 'index'? What is the definition of '>' (if it's different from the standard 'greater than' relation on natural numbers)? What is the definition of '<>'? But please do not give me yet more definitions that don't eventually lead to an endpoint with primitives only. Anyway, please, please, please, why don't you just learn something about mathematical logic so that you'd know how to formulate a definition. A definition does not have a free variable in the definiens that is not in the definiendum. You shouldn't even need mathematical logic for that; you instincts alone should warn you. MoeBlee
From: MoeBlee on 19 Sep 2006 13:54 Tony Orlow wrote: > You are wrong. If I assert that S = proper_subset(T) -> |S|<|T|, That wasn't the verbiage to which I responded. Your formula is messed up anyway ('=' is incorrect there, as is '(T)'). It should be: S is a proper subset of T -> |S|<|T|. Of course you can make that an axiom - if you first gave a logistic system, primitives, definitions of 'proper subset', '| |', and '<', or let them be primitives with (if you want to get some mathematics brewing) additional axioms for them. MoeBlee
From: MoeBlee on 19 Sep 2006 14:02 Tony Orlow wrote: > If you are so well-acquainted with these notions, could you be troubled > to define a distinction between them? Oh, please. They are well known concepts. I don't offer any explication beyond what you would find in any general presentation of the subject, just as I don't offer any special explication of such concepts as 'objective'/'subjective', or 'a priori'/'a posteriori', or hundreds of others. MoeBlee
From: Tony Orlow on 19 Sep 2006 14:24 Randy Poe wrote: > Han.deBruijn(a)DTO.TUDelft.NL wrote: >> Mike Kelly wrote: >> >>> Infinite natural numbers. Tish and tosh. Good luck explaining that idea >>> to schoolkids. >> Look who is talking. Good luck explaining alpha_0 to schoolkids. > > I think I was 10 when I saw the proof that the rationals are > countable, and first saw the notation "aleph_0". I don't remember > having a problem with it. > > - Randy > Perhaps you blocked it out. I know that, as soon as I was presented with the "proof" that there are as many evens as all naturals, even though that's only half of them, I detected a logical contradiction. The "equivalence" between the dense set of rationals and the sparse set of naturals, on the real line, only clinched it for me. It's simply unconscionable. The theory is wrong. As George Boole, who laid the groundwork for your automatic algebraic verification of logical Systems, the system by which they are established as being "internally consistent", said in his seminal work, "An Investigation Into The Laws Of Thought", "Let it be considered whether in any science, viewed either as a system of truth or as the foundation of a practical art, there can properly be any other test of the completeness and the fundamental character of its laws, than the completeness of its system of derived truths, and the generality of the methods which its serves to establish." Now, the general rule that adding more unique elements makes a set larger, that is, increases its measure of "size", is clearly violated by the "general" rules of set theory. If it serves as a foundation for all of mathematics, then how can it contradict any portion thereof? :) Tony
From: Tony Orlow on 19 Sep 2006 14:52
Mike Kelly wrote: > Han.deBruijn(a)DTO.TUDelft.NL wrote: >> Mike Kelly wrote: >> >>> Infinite natural numbers. Tish and tosh. Good luck explaining that idea >>> to schoolkids. >> Look who is talking. Good luck explaining alpha_0 to schoolkids. >> >> Han de Bruijn > > Sure, the theory of infinite cardinals is beyond (most)schoolkids. But > this is a bad analogy, because school kids don't need to know about > cardinals but they do need to know how to work with natural numbers. My > point, if you really missed it, was that Tony's ideas of "infinite > natural numbers" don't match up to our "naive" or "intuitive" idea of > what numbers should be - how we were taught to do arithmetic in school. > I for one don't understand what the hell an "infinite natural number" > is. And yet supposedly the advantage of his ideas are that they're more > intuitive than a standard formal treatment. > Hi Mike :) How are you? A natural number is a whole number. That may include both positive and negative whole numbers, because we can count equally well up or down. That means we take one unit at a time. We can treat 1 as 0 and vice versa, if we wish. They're only symbols. So, we can define increment and decrement as "find the rightmost this and invert rightward form there", or the "rightmost that". Or, m and w, if you recall. Math is all about the relationship between measure and symbolic representation. Can we easily decrement ...1111? Uh, ...1110. Increment? Well, that depends on whether we let the carry slide to oblivion, which depends on whether we are even allowing negative values. If ....111 is -1, then the successor is 0, or ...000, so we let it slide, that is, let the 1 bit fall off the end, and wrap from negative to positive. In this case the "largest positive finite" wold be something like "100...", with the leftmost 1 in the last finite position. But, that can't happen. So, oo can't really be represented in signed notation. Maybe that's honest. If ...111 is sum(n=0 -> w: 2^n), well, it's "the largest finite". You have no means to distinguish the largest finite value from the value of the greatest finite binary string, yet there is a power set relationship there, almost. :) Tony |