Any proof depends on the axioms. In some axiomatic systems, 1 + 1 = 2 is an axiom. In others, it indeed requires a proof. In Peano arithmetic, for example, it requires the application of two axioms of addition plus some equality reasoning.
Hey there, good day. Just wondering if we can prove this in a more complicated way. Our teacher once asked this in our class and I responded "Sir, we should not prove the obvious!" But i guess he wasn't contented. So, how should we attack this? that is, if there are any way. Thank you
Any proof depends on the axioms. In some axiomatic systems, 1 + 1 = 2 is an axiom. In others, it indeed requires a proof. In Peano arithmetic, for example, it requires the application of two axioms of addition plus some equality reasoning.
Using axioms as in the link above, S(0) + S(0) = S(S(0) + 0) = S(S(0)). The first equality as well as S(0) + 0 = S(0) are instantiations of axioms. Deriving S(S(0) + 0) = S(S(0)) from S(0) + 0 = S(0) requires the equality axiom x = y -> f(x) = f(y) for any functional symbol f. This axiom is a part of any first-order theory with equality, of which Peano arithmetic is an example.
1+1 can be 10 too, for example in binary system. In a number system based on 3 1+1=2 and 1+2=10 and so on. It's somewhat about the base of the number system you used or even the nature, for example roman number system is different in principle as well.
Though i am still figuring out how to work well with these unary functions, at least now i know how to begin my first step. all thanks go to you sir emakarov and sir zikcau25.
Before we can prove anything, we must accept some basic axioms. If your set of axioms includes 1+1=2, you are done. So, if you are looking for something "more complicated", your next step would be to create a set of basic axioms that does not include that sum, then prove that sum still follows as a logical conclusion. You need a definition for each symbol of your statement. So, what do the symbols 1,+,=, and 2 mean? For example, I could define the symbol 1 to be a boolean value FALSE, the + symbol to mean "is logically equivalent to", the = sign to mean the AND logical operator and the 2 to be the boolean value TRUE. Next, I can define an order that statement must evaluated. The order I define is that the = operator must be evaluated before the + symbol. So, we must show that 1=2 (which translates to FALSE AND TRUE) is always false. This requires the basic axioms of logic, so I will assume all of them. So, 1+1=2 means FALSE is logically equivalent to (FALSE AND TRUE). Then I can show using truth tables that this statement is TRUE.
Since those definitions for the symbols are vastly different from their standard definitions, they hold little practical value. While we could use them to prove the statement is true, they would not be usable outside the context of this one proof. The rest of the world uses standard definitions for each of those symbols. The symbols 1 and 2 represent numbers, the symbol + is a binary operation defined on a set of numbers, and the = symbol is a relation, which is a subset of the product of the set of numbers containing 1 and 2 with itself. Let's start by considering 1 and 2. Because I like the way it states it, I am copying the following from another website called cut-the-knot.org:
That list of kinds of numbers is by no means exhaustive. So, what kind of numbers are we looking for? How will we define them? Since we only care about two numbers for this proof, do we need to define an entire system of numbers? A truly axiomatic approach might require adopting a set theory. Next, we may have to prove that our set theory and number system even allows our expression 1+1=2. Once we have our set of numbers that contains both 1 and 2, we need to define an addition operation. Addition works in different ways in different systems. We may have to prove that our addition is well-defined. For example, in the finite field , since , so we may have to prove that our chosen set of numbers is closed under the addition we define. Since does not contain 2 in the first place, we obviously would not choose its addition operation, and I used it merely as an illustrative example. Finally, the = symbol needs a definition. To describe it more generically, it represents an equivalence relation. To make sure we have more to prove, perhaps we want the context of this equivalence relation to be on the collection of all arithmetic expressions. Now, you must define this collection and explain how to evaluate whether two of its elements are equivalent or not. Then you must prove that is an arithmetic expression. Then you must prove that is an arithmetic expression. I can then go back and decide that I want to have to prove even more. You could make your life's work the proof of that expression, yet no matter how atomic you make your initial axioms, your entire proof will still rely on those assumptions. Without them, you have no foundation for proof.To paraphrase Albert Einstein, a number in and by itself has no significance and only deserves the designation of number by virtue of its being a member of a group of objects with some shared characteristics. The most common characteristic of numbers is that they can be added and multiplied to produce other numbers in their group. However, not all objects that can be added or multiplied are designated as numbers.
As a matter of fact, there are many different kinds of numbers.
- rational and irrational
- real and complex
- imaginary
- algebraic and transcendental
- perfect
- surreal
- hyperreal numbers
- square and triangular numbers
Sir SlipEternal, i am still confused. How should i prove that 1 + 1 is an arithmetic expression? In your statement "so we may have to prove that our chosen set of numbers is closed under the addition we define," does that mean that i only need to add another digit on my set (which is 2) in order for my closure property under addition to be true?
To prove that 1+1 is an arithmetic expression, you first need to define what an arithmetic expression is. There are many ways to do this. One possibility is to use a free group generated by the elements of your set of numbers and a chosen set of arithmetic operators. Then, you would need to define some rules for selecting a subset of this free group that will contain only arithmetic expressions (possibly the entire free group, depending on how you will choose to evaluate your expressions). Depending on how you want your collection of arithmetic expressions set up, you may want to define an equivalence relation, then a function from your subset of the free group to a new set that take all equivalent arithmetic expressions (equivalence classes) to single elements of your new set (if you study abstract algebra, this notion will become more clear). Once you have what you consider your final collection of arithmetic expressions, you need to show that it contains the specific arithmetic expressions you are interested in: namely 1+1 and 2.
Yes, the set of numbers with the addition defined by 1+1 = 2, 1 + 2 = 1, 2 + 1 = 1, 2+2 = 2 obviously lets 1+1=2. But, this is not the standard addition that we are used to. So, why would we want this addition? If it is useless outside of this context, then we have no need for it. So, we may want to show that this addition is "similar enough" to standard integer addition to be a worthwhile operator. For instance, we may want to show that this addition is associative and/or commutative. Given any finite set of integers, we can show that standard integer addition is not closed over our finite set. So, perhaps we want to start with an infinite set. Again, this goes back to choosing an appropriate number system for the context under which we are trying to prove the statement 1+1=2.
While the responses are excellent, they are more what the OP wanted to learn for her needs. Perhaps the simplist presentation of Peano axioms are expressed by Bertrand Russell in his Introduction to Mathematical Philosophy, starting page 8 in the pdf file here http://people.umass.edu/klement/imp/imp-ebk.pdf which is manageable at introductory level of abstract mathematics. I read this book in French years ago. Finally I found it online in English.