Infinity…… Infinite? Or Finite?

<BLOCKQUOTE id=quote><font size=2 face="Verdana, Arial, Helvetica" id=quote>quote:<hr height=1 noshade id=quote>In order for infinity to exist does it require an initial starting point? If there were a beginning point would that point also serve as the end making what appeared to be infinite now finite?<hr height=1 noshade id=quote></BLOCKQUOTE id=quote></font id=quote><font face="Verdana, Arial, Helvetica" size=2 id=quote>

The classic treatise on the nature of infinities and how to treat them is Gamow's "One, two, three...infinity". Given your interest in this subject, I highly recommend a read. It will revolutionize your understandings. -|Tom|-

<BLOCKQUOTE id=quote><font size=2 face="Verdana, Arial, Helvetica" id=quote>quote:<hr height=1 noshade id=quote>Would you care to share your interpretation of Gamow’s work in the context of the question?<hr height=1 noshade id=quote></BLOCKQUOTE id=quote></font id=quote><font face="Verdana, Arial, Helvetica" size=2 id=quote>

The book explains the whole process of dealing with infinities. I recommended it because it will answer most of your questions all at once, not just the one on the table.

<BLOCKQUOTE id=quote><font size=2 face="Verdana, Arial, Helvetica" id=quote>quote:<hr height=1 noshade id=quote>In order for infinity to exist does it require an initial starting point?<hr height=1 noshade id=quote></BLOCKQUOTE id=quote></font id=quote><font face="Verdana, Arial, Helvetica" size=2 id=quote>

No. A straight line (as opposed to a line segment) extends to infinity and has no starting or ending point. Nor do its two ends ever meet.

<BLOCKQUOTE id=quote><font size=2 face="Verdana, Arial, Helvetica" id=quote>quote:<hr height=1 noshade id=quote>If there were a beginning point would that point also serve as the end making what appeared to be infinite now finite?<hr height=1 noshade id=quote></BLOCKQUOTE id=quote></font id=quote><font face="Verdana, Arial, Helvetica" size=2 id=quote>

Even a finite whole, as in a line segment, can be composed of an infinite number of parts (e.g., mathematical points). There are, for example, an infinite number of rational numbers between zero and one. The reciprocals of all the integers are an infinite subset of these. Gamow shows that, while all subsets of rational numbers have the lowest order of infinity, the number of irrational numbers is a higher order of infinity.

I previously avoided answering your specific questions because, without a background on how to deal with infinities, these answers amy simply lead to new questions, and so on to infinity. <img src=icon_smile.gif border=0 align=middle> -|Tom|-

If I may squeeze in here and note that G. Gamow was (I think) the creator of the Big Bang theory (A Ukrainian defector heavily influenced by creationists views), and further ask, how can his ideas fit in the Meta model, which ---if I understand correctly?--- rejects the Big Bang theory?

A deeper philosophical question is whether mathematical objects have any direct relation to the physical world.

<BLOCKQUOTE id=quote><font size=2 face="Verdana, Arial, Helvetica" id=quote>quote:<hr height=1 noshade id=quote>[makis]: note that G. Gamow was (I think) the creator of the Big Bang theory (A Ukrainian defector heavily influenced by creationists views), and further ask, how can his ideas fit in the Meta model, which ---if I understand correctly?--- rejects the Big Bang theory?<hr height=1 noshade id=quote></BLOCKQUOTE id=quote></font id=quote><font face="Verdana, Arial, Helvetica" size=2 id=quote>

The Big Bang was first formulated shortly after Hubble discovered the redshift relation in the late 1920s. (Of course, it did not acquire that name until Hoyle used it derisively in the 1950s.) Gamow is sometimes credited with the prediction of the cosmic background radiation as a fireball remnant of the Big Bang.

Gamow's views on math in general and infinities in particular are relatively isolated from his religious or cosmological views. They had a strong influence on me in my high school years, and ultimately did play a role in formulating the Meta Model. Infinity comes into it when trying to resolve Zeno's paradoxes. (See chapter 1 of <i>Dark Matter...</i>.)

<BLOCKQUOTE id=quote><font size=2 face="Verdana, Arial, Helvetica" id=quote>quote:<hr height=1 noshade id=quote>A deeper philosophical question is whether mathematical objects have any direct relation to the physical world.<hr height=1 noshade id=quote></BLOCKQUOTE id=quote></font id=quote><font face="Verdana, Arial, Helvetica" size=2 id=quote>

Mathematical objects are useful for describing the physical world. It remains a wonder of our times that I can sit here in an office, take mathematical formulations for the motions of celestial bodies, and predict future eclipses, occultations, meteor storms, and other phenomena with excellent precision and reliability.

In the case in point, Gamow's insights into dealing with infinities ("set up one-to-one correspondences") help us to understand how to answer Zeno's objection that one can never cross a street because he/she must first go half way, then half the remaining distance, and so on for an infinite number of steps without ever getting there. -|Tom|-

<BLOCKQUOTE id=quote><font size=2 face="Verdana, Arial, Helvetica" id=quote>quote:<hr height=1 noshade id=quote>Where on the line do you chose which direction you are going to follow? Where do you enter the line?<hr height=1 noshade id=quote></BLOCKQUOTE id=quote></font id=quote><font face="Verdana, Arial, Helvetica" size=2 id=quote>

Your question assumes that I have some perception of the line, or some interaction with it. Why should that be necessary? The line exists independent my my existence or anyone's perception of it. If it merely exists, no point on it is any different than any other.

Ironically, you said to makis: "This is a great philosophical question but please keep the original post of this thread to a mathmatical and scientific context." Are you sure you did not turn this discussion philosophical yourself by assuming that an observer is needed for reality to exist? -|Tom|-

<BLOCKQUOTE id=quote><font size=2 face="Verdana, Arial, Helvetica" id=quote>quote:<hr height=1 noshade id=quote>From a mathematical point of view, how can *you* enter the equation without having a starting point? [emphasis added]<hr height=1 noshade id=quote></BLOCKQUOTE id=quote></font id=quote><font face="Verdana, Arial, Helvetica" size=2 id=quote>

Why does there have to be a *you* at all? I gave a mathematical answer that is observer independent. The equation for a straight line (e.g., y=x) has no starting point. Only observers need starting points because observers are finite, limited beings. Mathematical lines are infinite, and all points on them are equivalent.

You have my answer, but seem to be trying to push in some predetermined direction, perhaps toward an answer you prefer better. Part of learning to be a good scientist is to recognize personal biases when they pop up, and prevent them from interfering in your quest for truth and understanding. -|Tom|-