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This is not a problem in math, but one in word meanings, i.e. check your dictionary. 'Infinite' literally means 'without a boundary' It's something without measure. Eg. the natural/counting numbers can be constructed using a Peano type rule. The set N containing them has no end, more correctly 'open ended'. Therefore there is no largest integer. Zero '0' or {} is the empty container. Math is the manipulation of the elements within the contaner. Any attempt to apply the rules to the container is nonsense.
The dictionary is the very last place anyone should ever go for a technical definition. If we apply mathematical definitions to your claims, every one of them is easily falsified. Math is full of counterexamples. The closed unit interval [0,1] in the reals has a boundary but is an infinite set. There are infinite sets with boundaries, infinite sets without boundaries, and infinite sets that contain some of their boundary points but not others. This is part of the subject of topology. The measure of the unit interval in the reals is 1, even though the unit interval is an infinite set. the measure of the interval between 1 and 3 is 2. This is just the distance formula from high school math. Plenty of infinite sets have a specific, finite measure. This is part of measure theory, an important branch of math. For that matter, even an unbounded set can have finite measure. For example the area under of the famous bell curve (the Gaussian probability distribution) from \(- \infty\) to \(\infty\), is \(\sqrt{\pi}\). This is easily proven in multivariable calculus. If the area weren't finite (hence scalable to 1), probability theory would not work. In the usual ordering. If you reorder the counting numbers with 0 stuck at the end, so that it's 1, 2, 3, 4, 5, ..., 0, then that's the exact same infinite set but with a largest element. Having a smallest or largest element is a function of the particular ordering we impose on a set; and it has nothing to do with how many elements are in the set. The same set may have a largest element or it may not, depending on how we choose to order it. See the discussion of ordinals a few posts back. That doesn't make much sense to me. I don't follow what you're trying to say. If the empty set is the empty container, which it is only in a vague, intuitive sense [since "container' is not a mathematical term in this context] then what does it mean to say that math is the manipulation of elements within the container? You just pointed out that there ARE NO ELEMENTS within the empty container. Nor is it true that math is about manipulating the elements within sets. A lot of modern math dispenses with elements entirely, and only cares about the relationships between and among various collections. And for that matter, a lot of ancient math doesn't care about elements in sets. You wouldn't say that Euclidean geometry is about manipulating elements within sets. And for that matter, set theory in its modern form is only about a century old. Before that, people did math but there were no sets. Well I hope my mathematical quibbles aren't too far off the mark relative to the general discussion. When people make incorrect statements about mathematical objects, my fingertips just can't help typing away.
someguy1; The dictionary includes references within the context of math. Math like any language, uses terms that are by definition. If posters understood the correct meaning of 'infinite', there would be no use of it as a number. It's a condition/relation, not a quantity. The unit interval U is finite and measurable. When defined as a continuum, the set of elements R (different from U) is without bounds. You're mixing apples and oranges. And what number would precede 0? Rearranging the elements of a set does not change the contents or the values. Only finite sets can contain a minimum or maximum value (because there are boundaries). If you tried to insert zero at various positions in an infinite line of people, each would tell you where to go, 'to the end of the line and take your zero with you'. Many generations would pass, but no one would ever see the headline "man discovers end of infinite line". Why not? An infinite integer is greater than any integer you can imagine. Rephrased, you can't imagine an infinite integer. The human mind does not have the capacity to do so, and no experience with instances of the idea. Even if humans were immortal, their lives would still be finite, since they have a beginning. With the Peano algorithm, successor of n is n+1, produces larger integers, but never a largest integer. The comparative and superlative case for 'large', grammar not math. That IS the point. Zero as a placeholder means there are no units in its location. Dictionary meaning: without quantity or magnitude. The container is typically not the same as the elements it contains, nor has the same properties (with the exception when the members are 'containers'), in the real world and the math world. Another reason why the mathematical operations shouldn't apply to the container. Supposedly a prominent mathematician defined mathematics as "a manipulation of symbols". It's the shortest, most concise description I've ever seen. Numbers are abstract representations of things, which cannot be processed easily or efficiently. In banking money isn't literally moved across the country, only the accounting data is transferred. From the beginning of civilization, people have counted, which requires sets/collections. Shepherds made a 1-1 correspondence with pebbles and sheep. When they returned, they would know if there were any losses. Any monetary transactions used some form of currency which represented the value of goods traded.
Part 1 (broken up due to forum software limitations) ==== You can't depend on the dictionary for technical definitions. And those definitions are to be found in textbooks and scientific papers. Not in the dictionary. Well that's just incorrect. In 1874 Georg Cantor initiated the study of transfinite ordinals and cardinals, giving those objects the status of numbers that can be added and multiplied. Treating infinite quantities as specific numbers having algebraic and order properties has been part of math for 140 years. https://en.wikipedia.org/wiki/Georg_Cantor https://en.wikipedia.org/wiki/Cardinal_number https://en.wikipedia.org/wiki/Ordinal_number The unit interval \(\{x \in \mathbb R : 0 < x < 1\}\) has uncountable cardinality. It's infinite. I can not imagine how you can possibly claim it's finite. It's a measurable set in the sense of measure theory, and its measure is 1. It's an infinite set having finite measure. https://en.wikipedia.org/wiki/Measure_(mathematics) It's true that \(\mathbb R\) is an unbounded set. But the unit interval is a bounded set. None of its elements are greater than 1. The closed unit interval \(\{x \in \mathbb R : 0 \leq x \leq 1\}\) contains its boundary points; whereas the open unit interval \(\{x \in \mathbb R : 0 < x < 1\}\) does not. The concepts of boundary point and boundedness pertain to the subject of general topology. These concepts have nothing to do with cardinality. You can have infinite sets that are bounded (such as the unit interval) or unbounded (such as the reals). https://en.wikipedia.org/wiki/General_topology I'm simply relating the state of the art in contemporary math regarding the treatment of concepts such as transfinite numbers and topological boundedness. Correct. Rearranging the elements of a set doesn't change the elements at all. But it does change the order of the elements. If we define a new order relation on the natural numbers \(1, 2, 3, 4, \dots, 0\) then these are the exact same elements as before. But the order is different. In our new order, the set of counting numbers has a largest element. This shows that the concept of cardinality, which refers to "how many," is very different than the concept of ordinality, which is about order. See the Wiki pages on cardinal and ordinal numbers I linked earlier. Of course this is not true. The closed unit interval is an uncountably infinite set that contains its minimum and maximum, namely 0 and 1 respectively. The natural numbers in "funny order" \(1, 2, 3, 4, \dots, 0\) is an infinite set that contains its minimum and maximum, 1 and 0 respectively. Remember, "how many?" and "what order are they in?" are distinct questions. One refers to cardinality and the other to ordinality. This math is 140 years old at this point and universally accepted. On the contrary. If I have a line of school kids I can tell them to line up by height, by weight, by alphabetical order of their last name, by reverse alpha order on their first name, by test scores, etc. Given any set I can order it in many ways. With infinite sets, we can reorder it so that it has a completely different order type, or ordinal number. Again, this is 140 years old and universally accepted in math. You don't have to accept math, but you do at least have to be honest and admit that you are officially rejecting math. I don't follow this at all. The two point compactification of the real line, also known as the extended real number system, is the set \([-\infty, \infty]\). It's introduced to students of freshman calculus. And believe me if freshman calc students can understand it, anyone can. https://en.wikipedia.org/wiki/Compactification_(mathematics) https://en.wikipedia.org/wiki/Extended_real_number_line Integers by definition are finite. There are no infinite integers in standard math. However we can certainly define infinite integers, or hyperintegers. This is done in nonstandard analysis. These numbers were discovered in 1948 and are of interest in the teaching of calculus and also in mathematical logic and topology. https://en.wikipedia.org/wiki/Hyperinteger https://en.wikipedia.org/wiki/Non-standard_analysis Continued ...
Part 2 ==== Mathematics routinely deals with infinity in various guises. From the extended reals of freshman calculus to the far reaches of large cardinals in set theory, mathematicians deal with infinity every day. https://en.wikipedia.org/wiki/Large_cardinal If you are arguing a finitist or ultrafinitist perspective, that's an interesting philosophy. But to be intellectually honest you have to say, "I don't believe in infinite sets. I'm a finitist or an ultrafinitist (as the case may be)." That way at least you would be making sense. But to make the statements you're making without clarifying your finitist or ultrafinitist stance is intellectually dishonest. It's confusing to readers who don't realize you're rejecting standard math. The definitions are: A finitist rejects completed infinite sets. An ultrafinitist rejects sufficiently large finite sets. To say we can't contemplate infinity because our lives are finite is an ultrafinitist argument. For example: A finitist accepts that there are infinitely many counting numbers as given by the Peano axioms. But the finitists deny that there is a completed set of these numbers. An ultrafinitist agrees that small positive integers like 3, 17, and 45454 exist; but they question the legitimacy of mathematical expressions such as \(2^{2^{2^{2^{2}}}}\), which is far larger than the number of atoms in the universe. https://en.wikipedia.org/wiki/Finitism https://en.wikipedia.org/wiki/Ultrafinitism Well, yes and no. The standard interpretation of the Peano axioms does not allow for infinitely large integers. But there are nonstandard models of the first-order Peano axioms that contain infinitely small and infinitely large integers. These are subtle issues in mathematical logic. https://en.wikipedia.org/wiki/Non-standard_model_of_arithmetic The dictionary is a terrible place to go looking for technical knowledge. In fact the quantity of elements in the empty set is 0. That's a particular quantity. We can add and multiply sets, as in the disjoint union and Cartesian product. That's one of the achievements of 20th century math: to be able to do algebra on collective objects such as sets. It's hard to know what that means. "Supposedly" some unidentified person said something? When I compose a forum post, aren't I manipulating symbols? Chemists manipulate symbols, as do physicists, economists, linguists, and novelists. Are all those activities now to be defined as math? But it gets worse. The formalists believed that math was nothing more than the manipulation of symbols according to formal rules. But in 1931, Gödel showed that mathematical truth can NOT be reduced to formal manipulation of symbols! That's an incredible breakthrough that destroyed the formalist dream. https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathematics) https://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems Short and concise, to be sure. But dead wrong on multiple grounds as I noted. The definition fails to distinguish between mathematics and novel writing; and it turns out that mathematical truth can NOT be reduced to formal manipulation of symbols. The formalist school of mathematical philosophy was destroyed in 1931. Indeed, math is abstract. But if you understand this, why would you earlier claim that we can't treat infinity mathematically because human lifespans are finite? Make up your mind. If math is abstract it can do anything it wants, unconstrained by physical limitations. Oh my. You're wrong about money too. If I'm in California and I transfer ten bucks to my friend in New York, I've moved money across country. These days money literally IS nothing more than bits in a computer. When your paycheck is electronically deposited to your checking account, where is that money? It's just bits flying around cyberspace. Not many people get paid or do business in cash any more. I hope this isn't news to you. Modern money moves from one place to another electronically, by altering entries in ledgers. This is not only true of cryptocurrencies, it's true of conventional fiat money. Have you heard of quantitative easing (QE)? A central bank creates money in its ledger, and lends the money to a bank. The bank turns around and uses that money to buy the bonds issued by the government associated with that bank. In this way the government can vastly increase the money supply yet keep interest rates low; a trick that's responsible for the massive "everything bubble" in stocks, bonds, and real estate that we see today. Modern money no longer consists of tangible things. And yes, it's regularly "moved" around the globe electronically. By this logic, quantum physics existed in ancient times because there were people and the people were made of atoms and quarks that obeyed quantum physics. Of course there were collections in the old days, but set theory as a formal discipline has only been around since 1874, and in its current form since as recently as the 1920's. You are failing to distinguish between reality, which is what it is, and human intellectual achievement, which is historically contingent. When I speak of set theory of course I mean the formal mathematical discipline of set theory. You're confusing the distinction between math/physics in reality, versus their formal existence as academic disciplines. Quantum physics has been around since the big bang, but was only discovered formally in the early 1900's. Likewise collections have always been around, but set theory is a relatively recent development in human intellectual activity. Which has what to do with the mathematics of infinity? You are being deliberately disingenuous. By your logic everything we know has always been known. That's not true. Set theory as a formal discipline dates from Cantor's 1874 paper, regardless of the fact that there were collection of things long before that. To sum up, you seem to be arguing from some sort of finitist or ultrafinitist perspective, but without acknowledging that you're doing so. It's perfectly legitimate to hold these philosophical perspectives. But it's confusing and a bit disingenuous to argue from these perspectives without making your perspective clear. Because then you just end up denying the entirety of modern math without explaining why.
I am aware of sometimes not making my thoughts clear in posting. A Peano type algorithm provides the potential to form integers of any desired magnitude. When formed they become real manifestations. The concept of 'infinite' resides in the mind only, and has no corresponding physical representation. It's easy to write 10^6 light yr, but no one can imagine it since it's so far removed from human experience. In physics, a theory may predict a particle x, but unless it is discovered, it remains hypothetical or non existent. No one will ever see an infinite list of integers, since by definition it has no end. No one will ever see the value of pi, only an approximation. What does that mean? Large compared to what? You can classify me in whatever group you like. phyti said: ↑ "That IS the point. Zero as a placeholder means there are no units in its location. Dictionary meaning: without quantity or magnitude." Doesn't my quote say the same thing as your quote? Isn't 0 in number theory the same as {} in set theory? It's not a quantity but an absence of quantity, just as dry is an absence of moisture. The empty set {} has no members!A good dictionary can provide an accurate definition of a word within a given field of study, and the word origin which clarifies the meaning. I don't argue against sets, just the role of {}.
You expressed your position perfectly clearly. It's consistent with the philosophical doctrine of ultrafinitism. It's like if I came to a philosophy forum and I said, "I believe there are two types of things: the physical, and the mental. Both of these things exist and they are different." If I said that, someone would point out that I'm simply expressing what's known as Cartesian dualism. If I then said, "Oh no you're just labelling me," that would be inaccurate. I may not have studied Cartesian dualism formally, but that is simply the name for the belief that there are two kinds of substance: mind and matter. Likewise, if you believe that a number can be said to exist if it can be instantiated in the real world; and otherwise it can't be said to exist; that's pretty close to ultrafinitism. It's a perfectly sensible view, its only drawback is that it's hard to get science off the ground without infinitary math. [That in itself has philosophical implications. WHY is physics based on infinitary math when we all agree that there are no actual infinities in the real world? Conversation for another time, and an interesting one]. That's a nonstandard view of what the Peano axioms (not algorithm) are. When you take the successor of a number, that does not bring the successor into existence or make it a "real manifestation." The Peano axioms just show that we can formalize our intuition of the natural numbers by writing down a small number of perfectly plausible axioms. Although the induction axiom is NOT perfectly plausible at all. How can you "keep going forever" in a finite universe? That's where the ultrafinitists come in. They cast doubt on the inductive axiom of Peano. Ok, perfectly agreed. Although I'd say that the number 3 resides in the mind only as well, but that's a side issue. The thesis you are making is this: A number can claim to "exist" if it can be instantiated in the real world. A number that "has no corresponding physical representation," as you say, does NOT have a claim to existence. That's ultrafinitism. You are saying that the number 3 exists because I can instantiate 3 in the real world, as in three chairs or three apples. But then what do we make of a number like \(2^{2^{2^{2^{2}}}}\)? This expression represents a number that is far larger than the number of atoms in the universe. The number denoted by this expression could never exist in the real world. In the standard philosophy of math, that number exists every bit as much as 3 does. To an ultrafinitist, it's not actually well-defined. The way ultrafinitsts put this is that "Exponentiation is not a total function." In other words exponentiation does not always give a sensible answer given arbitrary inputs. If we wanted to count upward by 1's till will got to my expression, how would we know exactly when to stop? It's a good question actually. If you are saying that numbers exist only to the extent that they can be instantiated in the real world, that is the position of the ultrafinitists. I hope you don't think I'm making a criticism. Ultrafinitism is a perfectly valid philosophy of math. It just doesn't happen to be useful in getting modern math off the ground, let alone physical science. Right. Just like ultrafinitism. But in standard math, a number exists when the axioms say it exists. Modern math accepts the axiom of infinity, so infinite sets exists. Of course nobody is saying that infinite sets have physical existence. Nor does math say that a set containing 3 elements has physical existence! Only that we can represent it within set theory. True. You're making an ultrafinitist argument. You're saying that an infinite set doesn't exist because we'll never see one in front of us. You are absolutely correct about that. That's why ultrafinitism is perfectly sensible. It's just not useful, because we can't do any modern math with it. Yes, that's also the point. Nobody will ever see any of the noncomputable numbers (these are the real numbers whose digits can not be cranked out by any algorithm). Nobody will ever see \(2^{2^{2^{2^{2}}}}\). If you accept the existence of these numbers because their abstract, mathematical existence can be proved from the standard axioms of math, that makes you a "standardist." I don't think there's a name for it. Perhaps "infinitarist." If you deny that a number exists if that number can't be represented in the world, you're an ultrafinitist. There's an intermediate position. A finitist believes in each of the counting numbers 1, 2, 3, 4, ... individually, but not all of them taken together as a set. In other words they accept the existence of the numbers given to us by the Peano axioms, but not the axiom of infinity. You said the Peano axioms don't give us infinite integers. I was agreeing with you. I'm only pointing out that you are expressing the perfectly legitimate beliefs of the philosophy of ultrafinitism. Just like if I say there are two substances, mind and matter, I'm expressing the beliefs of Cartesian dualism, even if I've never heard it put that way. Here you're using zero as a numeral in a positional system of notation. Regardless, if you are arguing against the existence of the number zero, again that's not wrong, it just puts your mathematical thinking back in the middle ages somewhere. Being modern doesn't make us right. I'm not saying you're wrong, only that you are rejecting almost the entirety of modern math. If you quoted from a math book that would be more compelling evidence. Yes, {} is the way we represent the number 0 within set theory. They are not the "same" in the sense that 0 is an abstract number, and {} is merely its representation or definition in the formal system of set theory. One is an abstract thing, the other its representation. It's a minor semantic point, but a mathematician would say, "Zero is not nothing!" For example the real number zero is a particular location on the number line. If you're at coordinate 0 or you're at coordinate 47, you're still somewhere. A real number is just a label for a location. Correct. It's the set of all the purple flying elephants. Well you have only quoted from "the dictionary" but you have not specified whether you are using a good dictionary or not. Regardless, if you want the technical meaning of a word, you need to refer to the technical literature and not a general purpose dictionary. I don't know why you're arguing this point. When you go to medical school they hand you an anatomy text, not the Oxford unabridged dictionary. Ah, you're arguing against the empty set. Well you're not alone. One can make many substantive philosophical criticisms of set theory, and the nonsensical notion of the "empty set" is surely one of them. I myself don't argue for the physical reality of the empty set. I tend to agree with you on this point. However, we DO have to accept the existence of the empty set within the formal system of set theory, because the existence of the empty set can be proved from the axioms of set theory. One need not accept set theory; but one does at least have to agree that within set theory, the empty set exists. There's another philosophical doctrine of interest, mathematical fictionalism, which says that our mathematical entities are fictions much as are the characters in a novel. https://plato.stanford.edu/entries/fictionalism-mathematics/
A number can be considered as distance between two points. This distance can be measured with a known distance between two fixed points. So number is a relative concept. If we measure all the distances with relative to the highest number, then the highest number can be \( 1 \) also.
I agree. Within the context of infinity, one can be considered the largest number. But then so can two, three and even + zero...
Not denying it is used, but not as a quantity, more as a logical tool as in the definition of a limit. You missed the point, being defining math in fundamental terms. Math is a language, with a set of symbols and a rigid syntax for forming expressions, which appeal to a smaller segment of society. The typical language has a set of symbols for forming expressions, but a relaxed syntax, which allows for more interesting reading to a larger segment of society. So in general there is a common activity of manipulating symbols. Chemists and physicists use math. The mathematician I quoted was only describing the application of math, not its ability to produce true statements. He most likely was aware of Godels proof. There seems to be an attitude that develops among intellectuals, that if they can solve some problems, they can solve all problems. That's a giant leap of faith. We need more Godels to provide a reality check. Not because of human lifespans (I allowed immortality). Infinite is not quantifiable. If the people responsible for introducing it into the world of math had researched its meaning, they might have reconsidered.. Read it again! People from past millenna have used numbers in some form, for various purposes, without the necessity of formal systems. You are distorting the meaning of my words. I'm just giving earlier generations credit for being creative without technology and formal systems of education. You should have me classified by now.
