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>This is clearly not true. Adding axioms (as long as they >don't introduce consistency) can only reduce the set of >models. Anything that satisfies the larger t

>This is clearly not true. Adding axioms (as long as they >don't introduce consistency) can only reduce the set of >models. Anything that satisfies the larger theory also satisfies the smaller theory. This is exactly why a group is a monoid is a semigroup. This is why fields are rings. In fact, this is why the rationals or the reals are fields: they satisfy the field axioms, and more (for example, they are ordered). Look at for example the set of real numbers. If you add the complex numbers to it, you can no longer speak of the real numbers. The real numbers form a subset of the complex numbers. The same thing happens if you add definitions to a field. You create something which has mayber has a field as a subset, but you can no longer speak of a field. What is "undefined"? First let's look at the logic itself: What is true ∨ undefined ? What is true ∧ undefined? Now at the theory: What is undefined + x ? What is 0 * undefined ? What is 0 = undefined ? etc.[0] Undefined is just undefined, no value, not a magic one, or just a random one, or one you prefer. Just void, the empty set. To understand it, look for example at the graph of y = sqrt(x), where x and y are real numbers. You see no points at negative x. Its undefined in the domain x < 0. Yes you can say at x < 0, y = 3, that is now my definition sqrt(), but then you no longer have the sqrt function mathematicians talk about. The same thing applies to fields and 1/x. >What is "undefined"? First let's look at the logic itself: >What is true ∨ undefined ? What is true ∧ undefined? Now at the theory: What is undefined + x ? What is 0 * undefined ? What is 0 = undefined ? etc.[0] Now you apply boolean logic to the result of calculations, numbers and non-numbers, not on statements. I can also ask what is 5 AND true? Is that true? The question doesn't make sense, so the answer is: depends on the programming language.

pron 8y ago
> Look at for example the set of real numbers. If you add the complex numbers to it, you can no longer speak of the real numbers. What does this have to do with adding axioms? You don't get the complex numbers from the real numbers by adding axioms. For example, the following is provable for the reals, but not for complex numbers: ∀x,y . x < y ∨ x > y ∨ x = y > You create something which has mayber has a field as a subset, but you can no longer speak of a field. No. The class of all fields are all objects satisfying the field axioms. If you add axioms, you get get a subset of those fields. Each and every one of them is a field. > Undefined is just undefined I'm not looking for handwaving. Formal mathematics is math that can be done mechanically. Write down the axioms for undefined. Again -- I'm not saying it hasn't been done, but it's certainly not part of standard first-order logic, and it is unnecessary. > To understand it, look for example at the graph I understand what undefined means in informal mathematics. Try to see what it means in formal mathematics. But the important point is, try to understand why it is unnecessary in formal mathematics in most cases. > Now you apply boolean logic to the result of calculations, numbers and non-numbers, not on statements. No. You can write 1/x < 5 ∨ x > 20. > The question doesn't make sense, so the answer is: depends on the programming language. There is no such thing as "doesn't make sense" in formal math, even though there is such a thing in informal math. That's the whole point. Either the expression is ill-formed, i.e. not in the language or a "syntax error", or it must make some sense. This is why it's useful and easy to say something is undefined in informal math, but not as useful and not as easy to do that in formal math.
- blablablerg 8y ago
  >What does this have to do with adding axioms? You don't get the complex numbers from the real numbers by adding axioms. For example, the following is provable for the reals, but not for complex numbers: > ∀x,y . x < y ∨ x > y ∨ x = y Since R is a subset of C, you can write C as R with additional axioms. See for example: http://www.math.mcgill.ca/gantumur/math249w15/numbers.pdf http://www.math.mcgill.ca/gantumur/math249w15/numbers.pdf >I'm not looking for handwaving. Formal mathematics is math that can be done mechanically. Write down the axioms for undefined. Again -- I'm not saying it hasn't been done, but it's certainly not part of standard first-order logic, and it is unnecessary. Undefined is just undefined, that is no handwaving, it is a primitive notion. See https://en.wikipedia.org/wiki/Primitive_notion https://en.wikipedia.org/wiki/Primitive_notion You are trying to define the undefined in a formal system. Undefined is just the absence of a definition. Not 0, 3 , 2pi > No. You can write 1/x < 5 ∨ x > 20. Yes, those are valid mathematical statements. 5 ∨ TRUE, or undefined V TRUE, I doubt it. > There is no such thing as "doesn't make sense" in formal math, even though there is such a thing in informal math. >That's the whole point. Either the expression is ill-formed, i.e. not in the language or a "syntax error", or it must make some sense. Well in language, in which I am corresponding with you, there is such a thing as 'makes no sense'. You are somehow trying to capture everything in your logical system only to try to prove that 1/0 = 0 is part of a field, which it isn't. I've made my point here, it was nice talking to you.
  - pron 8y ago
    > Since R is a subset of C, you can write C as R with additional axioms. That's not even remotely how it works. You may want to read up on logical theories and models[1]. > Well in language, in which I am corresponding with you, there is such a thing as 'makes no sense'. Yes, and for the same reason we can have such a thing as "undefined" (that means more than merely 'not specified') in informal mathematics -- because both English and informal math are informal. But we are talking about formal languages[2], which do not have such a thing. [1]: https://en.wikipedia.org/wiki/Theory_(mathematical_logic) https://en.wikipedia.org/wiki/Theory_(mathematical_logic), https://en.wikipedia.org/wiki/Structure_(mathematical_logic) https://en.wikipedia.org/wiki/Structure_(mathematical_logic) [2]: https://en.wikipedia.org/wiki/Formal_system https://en.wikipedia.org/wiki/Formal_system

pron 8y ago

> Look at for example the set of real numbers. If you add the complex numbers to it, you can no longer speak of the real numbers. What does this have to do with adding axioms? You don't get the complex numbers from the real numbers by adding axioms. For example, the following is provable for the reals, but not for complex numbers: ∀x,y . x < y ∨ x > y ∨ x = y > You create something which has mayber has a field as a subset, but you can no longer speak of a field. No. The class of all fields are all objects satisfying the field axioms. If you add axioms, you get get a subset of those fields. Each and every one of them is a field. > Undefined is just undefined I'm not looking for handwaving. Formal mathematics is math that can be done mechanically. Write down the axioms for undefined. Again -- I'm not saying it hasn't been done, but it's certainly not part of standard first-order logic, and it is unnecessary. > To understand it, look for example at the graph I understand what undefined means in informal mathematics. Try to see what it means in formal mathematics. But the important point is, try to understand why it is unnecessary in formal mathematics in most cases. > Now you apply boolean logic to the result of calculations, numbers and non-numbers, not on statements. No. You can write 1/x < 5 ∨ x > 20. > The question doesn't make sense, so the answer is: depends on the programming language. There is no such thing as "doesn't make sense" in formal math, even though there is such a thing in informal math. That's the whole point. Either the expression is ill-formed, i.e. not in the language or a "syntax error", or it must make some sense. This is why it's useful and easy to say something is undefined in informal math, but not as useful and not as easy to do that in formal math.

blablablerg 8y ago
>What does this have to do with adding axioms? You don't get the complex numbers from the real numbers by adding axioms. For example, the following is provable for the reals, but not for complex numbers: > ∀x,y . x < y ∨ x > y ∨ x = y Since R is a subset of C, you can write C as R with additional axioms. See for example: http://www.math.mcgill.ca/gantumur/math249w15/numbers.pdf http://www.math.mcgill.ca/gantumur/math249w15/numbers.pdf >I'm not looking for handwaving. Formal mathematics is math that can be done mechanically. Write down the axioms for undefined. Again -- I'm not saying it hasn't been done, but it's certainly not part of standard first-order logic, and it is unnecessary. Undefined is just undefined, that is no handwaving, it is a primitive notion. See https://en.wikipedia.org/wiki/Primitive_notion https://en.wikipedia.org/wiki/Primitive_notion You are trying to define the undefined in a formal system. Undefined is just the absence of a definition. Not 0, 3 , 2pi > No. You can write 1/x < 5 ∨ x > 20. Yes, those are valid mathematical statements. 5 ∨ TRUE, or undefined V TRUE, I doubt it. > There is no such thing as "doesn't make sense" in formal math, even though there is such a thing in informal math. >That's the whole point. Either the expression is ill-formed, i.e. not in the language or a "syntax error", or it must make some sense. Well in language, in which I am corresponding with you, there is such a thing as 'makes no sense'. You are somehow trying to capture everything in your logical system only to try to prove that 1/0 = 0 is part of a field, which it isn't. I've made my point here, it was nice talking to you.
- pron 8y ago
  > Since R is a subset of C, you can write C as R with additional axioms. That's not even remotely how it works. You may want to read up on logical theories and models[1]. > Well in language, in which I am corresponding with you, there is such a thing as 'makes no sense'. Yes, and for the same reason we can have such a thing as "undefined" (that means more than merely 'not specified') in informal mathematics -- because both English and informal math are informal. But we are talking about formal languages[2], which do not have such a thing. [1]: https://en.wikipedia.org/wiki/Theory_(mathematical_logic) https://en.wikipedia.org/wiki/Theory_(mathematical_logic), https://en.wikipedia.org/wiki/Structure_(mathematical_logic) https://en.wikipedia.org/wiki/Structure_(mathematical_logic) [2]: https://en.wikipedia.org/wiki/Formal_system https://en.wikipedia.org/wiki/Formal_system