TheDivineFool Posted April 3, 2015 Share Posted April 3, 2015 I'm just a novice in maths. So, I read this elementary book on math and my knowledge on the topic is summarized below... 1. Infinity is not a number. Infinity is to us is like googol to a calculator...overflow. It's an unquantifiable. In that sense, I always saw infinity to be closer to a quality such as color than say the number 4 or 5 etc. 2. Definition of infinity (using sets) Definition 1: A finite set is a set equivalent to the set {1, 2, 3,..., n}. An infinite set is NOT an finite set. Definition 2: An infinite set is a set that is equivalent to a proper subset of itself. A finite set is a set that is not infinite. Question: Which of these definitions is better? I find 1 better because it captures the essence of ''endlessness'' that defines infinity. Definition 2 is roundabout and more difficult for me. Can someone deepen my understanding of infinity. Thanks. Link to comment Share on other sites More sharing options...

studiot Posted April 3, 2015 Share Posted April 3, 2015 (edited) Well the key to understanding either definition is to understand what is meant by 'equivalent'. Stating "set A is equivalent to set B means that you can put each and every element of set A in one-to-one correspondence with an element of set B, with no elements 'left over' in either set A or set B. So definition 1 says that the elements of every finite set can be put into one to one correspondence with the set of intergers {1,2,3,4,.....n}, but that an infinite set is a set for which you cannot do this. Definition 2 says that you can chop out a chunk of an infinite set (posh words = proper subset) and put its members in one to one correspondence with the whole set, which is not possible for a finite set. For instance you can put every real number between -1 and +1 into one to one correspondence with every real number. Does this help? Edited April 4, 2015 by studiot 2 Link to comment Share on other sites More sharing options...

John Posted April 4, 2015 Share Posted April 4, 2015 Definition 1 is the one I've seen most often. Either definition is fine for the time being, though the second definition is only correct assuming the axiom of choice (which is, admittedly, pretty well-accepted these days). To explain further, the first part of definition 2 defines what is called a "Dedekind-infinite" set, and while every Dedekind-infinite set is infinite under Zermelo-Fraenkel set theory, we can prove the converse only if we invoke the AC (well, really just the axiom of countable choice, but eh). That is to say, without AC_{(ω)}, we cannot prove that every Dedekind-finite set is finite, so the second part of definition 2 fails. Link to comment Share on other sites More sharing options...

TheDivineFool Posted April 4, 2015 Author Share Posted April 4, 2015 Thanks both of you. Now I'd like to present an experiment that will, I hope, reveal the nature of infinity... Pick the odd one out from the following list: zero, 187, red, triangle, infinity Thanks. Link to comment Share on other sites More sharing options...

studiot Posted April 4, 2015 Share Posted April 4, 2015 (edited) If it looks like a duck and walks like a duck and quacks like a duck.................................... The definitions you posted in post1 are definitions by property. That is properties of the defined object are identified. The trouble with that sort of definition is that you can never be certain if your list is complete. However the more properties you can identify the more confidence you can have in your definition, as with the duck. Further in both post1 and with your riddle is only one property is identified. Negative properties ie "X is not ..." are really useless in definitions, although they can offer some intuitive feel. In your riddle list there is on one item written in decimal digits so, from that point of view it is the odd one out. As "is not..." might be that there are two words with eight letters so neither can be candidates from the property of number of letters. Edited April 4, 2015 by studiot Link to comment Share on other sites More sharing options...

TheDivineFool Posted April 4, 2015 Author Share Posted April 4, 2015 One comment that I've seen a lot on the www is that ''infinity is not a number''. With my limited understanding, this is why I think people say this... A number is the result of a counting process. {Apple, love, $} is of size 3 i.e. there are 3 elements in the set. Similarly {u, 4, red, dog, relativity} is of size 5 i.e. there are 5 elements. As you can see, to get a number that measures the quantity of something requires a beginning of the counting process and then the process must END. With an infinite set e.g. {1, 3, 5,...} we can see that the counting process has a beginning BUT it has no end. Since counting cannot end, there is no NUMBER we can assign to the infinite set of odd numbers. Therefore, infinity is not a number. Am I right? If I am, I find this strange. I mean here's a 'thing' which sits somewhere between a quantity (countable) and a quality (uncountable). That's why I presented the riddle. I feel infinity sits between 'red' and '145'. It is neither a number nor a non-number. In this sense, it has similarities to zero. Zero is an 'absence' of a quantity. Likewise infinity is the other end of the spectrum, a quantity that is unquantifiable. Both are 'singularities' where math breaks down. If it looks like a duck and walks like a duck and quacks like a duck.................................... The definitions you posted in post1 are definitions by property. That is properties of the defined object are identified. The trouble with that sort of definition is that you can never be certain if your list is complete. However the more properties you can identify the more confidence you can have in your definition, as with the duck. Further in both post1 and with your riddle is only one property is identified. Negative properties ie "X is not ..." are really useless in definitions, although they can offer some intuitive feel. In your riddle list there is on one item written in decimal digits so, from that point of view it is the odd one out. As "is not..." might be that there are two words with eight letters so neither can be candidates from the property of number of letters. I agree, a definition in negative is usually undesirable. But, these are not my definitions. These are some I picked up from a math book. Also, I think definition 1 (see my OP) is actually a mathematical translation of infinity as used in ordinary conversation which is, roughly speaking, ''the state of being endLESS''. So, you can see, 'infinity' is defined in the negative (WITHOUT end). This is exactly what definition 1 states, in math context. Link to comment Share on other sites More sharing options...

studiot Posted April 4, 2015 Share Posted April 4, 2015 With an infinite set e.g. {1, 3, 5,...} we can see that the counting process has a beginning BUT it has no end. Since counting cannot end, there is no NUMBER we can assign to the infinite set of odd numbers. Therefore, infinity is not a number. Am I right? In English maybe, but not in mathematics. Look up countable (denumerable is a posh word) and countably infinite. Link to comment Share on other sites More sharing options...

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