# Is this for philosophers, mathematicians, or logicians? But here goes: Given that the decimal places of pi continue to infinity, does this imply that somewhere in the sequence of numbers of pi there must be, for instance, a huge (and possibly infinite) number of the same number repeated? 77777777777777777777777777... , say? If Pi goes on forever, you might think it must be. After all, if you checked pi to the first googol decimal places you obviously would't find an infinite number of anything. Try a googlplex! Still nothing. But we haven't scratched the surface, even though the universe would have fizzled out by now. If pi's decimal places go on forever, there may be, (not just 77777777777777... or 1515151515151) but all of them, in all combinations, forever. After all, you only have to say "You've only checked a googolplex. There's still an infinite number to to check. The universe is long gone, but pi goes on and on." Philosophers, mathematicians, logicians, any ideas? Mark G.

There are two questions here that need to be distinguished: (1) Does the decimal expansion of pi contain a large but finite string of consecutive 7's--say, 1000 consecutive 7's? (2) Does it contain an infinite string of consecutive 7's? The second question is the easier one. The only way that pi could contain a string of infinitely many consecutive 7's is if all the digits are 7's from some point on. And, as William has pointed out, that can't happen because pi is irrational. But the first question is harder. The digits of pi "look random." Imagine a number whose digits are generated by some random process--for example, we might roll a 10-sided die repeatedly to generate the digits. One could compute the probability that a string of 1000 consecutive 7's appears in the first n digits of this number. For n =1000, this number would be extremely small--you'd have to roll 1000 consecutive 7's on your first 1000 rolls, and that's very unlikely. But as n increases, the probability increases, and...

# In the probability thread, multiple philosophers mention examples of zero-probability events that aren't necessarily "impossible" (like flipping an infinite number of heads in a row). Arriving at a probability of zero in these instances relies on saying that 1/infinity = 0. But this math seems misleading. Don't mathematicians rely on more precise language to avoid this paradoxical result, by saying that "the limit of 1/x as x approaches infinity = 0," rather than simply "1/x = 0"? I feel like there must be some way to distinguish (supposedly) zero-probability events that are actually possible and zero-probability events that are impossible. Thanks!

To answer this question, it may be helpful to say something about the mathematical formalism usually used in probability theory. The first step in applying probability theory to study some random process is to identify the set of all possible outcomes of the process, which is called the sample space . For example, in the case of an infinite sequence of coin flips, the sample space is the set of all infinite sequences of H's and T's (representing heads and tails). Probabilities are assigned to events , which are represented by subsets of the sample space. For example, in the case of an infinite sequence of coin flips, the set of all HT-sequences that start with H represents the event that the first coin flip was a heads, and (assuming the coin is fair) this event would have probability 1/2. The set of sequences that start with HT is a subset of the first one, and it represents the event that the first flip was heads and the second tails; it has probability 1/4. Now, consider some infinite HT...

# Consider a first-order axiomization of ZFC. The quantifiers range over all the sets. However, we can prove that (in ZFC) there is no set which contains all sets. Soooo.........how can we make a _model_ for ZFC? The first thing you do when you make a model for a set of axioms is specify a domain, which is a set of things which the quantifiers range over......this seems to be exactly what you can't do with ZFC. So what am I missing?

You are right that the first thing you do when you make a model for set theory is to specify a domain. You also have to specify an interpretation for the "is an element of" symbol. For example, you might specify that the domain is to be the set of natural numbers, and the symbol for "is an element of" is to be interpreted to mean "is less than". Of course, under this interpretation most of the axioms of set theory would come out false, so although this is a possible interpretation for the language of set theory, it is not a model of ZFC. But if ZFC is consistent, then it has a model whose domain is the set of natural numbers. In this model, the "is an element of" symbol would be interpreted as some relation on the natural numbers. (The existence of such a model follows from the Lowenheim-Skolem theorem, which says that if ZFC is consistent, then it has a countable model.) Now, you might object that this model is not the intended model, because our intention is that the variables...

# How much math should I know in order to delve really deeply into philosophy of mathematics? Must philosophers of mathematics be mathematicians, as well?

Philosophers of mathematics don't have to be mathematicians, but it would be helpful to know a fair amount of math. Here are some more specific suggestions: 1. You need to study enough math to appreciate the role of proofs in mathematics. Usually students don't see this until they get beyond calculus. Courses on analysis and abstract algebra would show you this side of mathematics. (Abstract algebra would also allow you to see the role of abstraction in math.) 2. There are ideas from logic, such as Godel's incompleteness theorems, that are important in philosophy of math, so you should study logic. 3. You should know about how some of the fundamental objects studied in mathematics are defined. For example, how are the real numbers defined? Usually mathematicians trace the ideas behind the definitions of these fundamental mathematical objects back to set theory, so it would be good to learn some set theory.

# What does it mean in mathematics for two things to be equal, or for two things to have the same "identity"? For example, because anything divided by zero is "undefined", can we say that 1/0 = 2/0? What about the relational database concept of "null" which is supposed to stand for "unknown"? In relational algebra, they say NULL is not equal to NULL, but doesn't that violate the law of identity that everything is equal to itself?

I think it is important to distinguish here between the meanings of expressions and the things that those expressions denote. Peter is right that the expressions "2+2" and "4" are different expressions, and they are not synonymous. But they both denote the same thing, namely the number 4. Now, in the equation "2+2=4", is the equal sign being used to express a relationship between meanings of expressions, or does it express a relationship between what is denoted by those expressions? (In other words, is this an intensional context or an extensional context?) I would say it expresses a relationship between what is denoted by the expressions, and the relationship is identity: what is denoted by the two sides of the equation is one and the same thing, namely the number 4. So I disagree with Peter's conclusion about what "=" means in mathematics. I would say "=" in mathematics means "is identical with". I would say that the situation here is very much like the situation in the sentence "The morning...

# I have been studying axiomatic set theory as a foundation of mathematics and am stuck on the definition of a relation as a subset of a Cartesian product. I have two problems. The first is that a large number of relations seem to be presupposed prior to this definition: the truth-functional relations of logic, for example, or the relations of set-membership and subset. Doesn't this make the definition circular? Second, in specifying which subset of the Cartesian product is intended, a polyadic predicate is usually invoked; but isn't a polyadic predicate a relation, thus giving a second circularity? Furthermore, these are vicious circles, not harmless ones.

All theorems about relations in axiomatic set theory are proven just from the axioms, using the rules of first-order logic. Thus, no facts about relations are presupposed in these proofs--at least, not if by "presupposed" you mean "used to justify a step in a proof." But perhaps this is not the sense of "presupposed" that you have in mind. Perhaps what you are thinking is that in order to really understand what's going on in the development of the theory of relations in axiomatic set theory, you have to have an intuitive understanding of what a relation is. Or perhaps you mean that no one would believe that the rules of first-order logic represent correct reasoning, or that the axioms of set theory are true statements about sets, if they weren't familiar with certain relations, such as the truth-functional relations of logic or the set-membership relation. You may be right about this. Is this a problem for the idea that axiomatic set theory is a foundation for mathematics? Not necessarily. ...

# Hello philosophers. I was just wondering about Gödel's Incompleteness Theorem. What exactly is it and does it limit what we are capable of knowing? I have no training in mathematics or formal logic so if you could reply in lay terms, I would appreciate that. Thanks, Tim.

Godel's Incompleteness Theorem is a theorem about formal axiomatic theories: theories in which there is a collection of axioms from which all of the theorems are deduced, and in which the theorems are deduced from these axioms by the application of rules of logic. It applies to a wide range of theories, but to start off it might be helpful to focus on one such theory, so let's consider Peano Arithmetic, often abbreviated PA. This is an axiomatic theory of the properties of addition and multiplication of the natural numbers 0, 1, 2, ... (Peano Arithmetic is named after Giuseppe Peano.) Godel's Incompleteness Theorem says that if PA is consistent--that is, if the axioms don't contradict each other--then there are statements about the arithmetic of the natural numbers that are neither provable nor disprovable from the PA axioms. Thus, the axioms are not powerful enough to settle every question of number theory. Now, you might think that all this shows is that Peano must have forgotten an axiom...

# To Whom it May Concern: Mathematical results are assumed to be precise. But how can mathematics be precise if results are rounded up or down? Don't such small incremental "roundings" add up to imprecision? So, in general, don't "roundings", in some way, betray the advertised precision of mathematics? Sincerely, Alexander

You're right, mathematical results will not be precise if they are rounded off -- which is why mathematicians usually don't round off their results. I think such rounding is much more common among people who are applying mathematics to real world problems than among mathematicians doing theoretical work. For example, consider the question: What is the height of an equilateral triangle whose sides have length 1? A mathematician would most likely say that the answer is sqrt(3)/2, which is exactly correct. But someone who needed the answer to this question in order to apply it to a real world situation might prefer to have a decimal value, and so they might round it off to 0.87.

# How can was say that a variable such as x exists as a number or at all in an equation when by using a variable we claim to know nothing of what "goes in there" to complete the equation?

I would add that it is important to distinguish between a variable and what the variable stands for. You ask how we can say that "a variable such as x exists as a number." I would say the variable is a letter, not a number, but it stands for a number. As Richard has explained, what number it stands for, or whether there is a single number that it stands for, may depend on the context.

# Is infinity a number or not and why?

I think it's worth pointing out that there are many different number systems, which mathematicians use for different purposes, so your question is really ambiguous. If you are interested in determining how many things of some particular kind there are, then the appropriate numbers to use are the cardinal numbers, and as Richard has explained, there are indeed infinite cardinal numbers. On the other hand, if you're a student in a calculus class, then the numbers you are using are probably the real numbers, and all of the real numbers are finite. Although the symbol for infinity is used in calculus, it is not used as a name for a real number.