proof of 1+1=2 with justification-thetnfinitymathematics

 

PROOF OF 1+1=2 WITH JUSTIFICATION

Dan Bdr Budha

Student, Central Department of Education, University Campus, Kirtipur, Nepal

E-mail: budhadanbdr@gmail.com

 

The proof of a mathematical proposition is a finite sequence of statements ending in the proposition, which satisfies the following property. Each statement is an axiom drawn from a previously stipulated set of axioms or is derived by a rule of inference from one or more statements occurring earlier in the sequence. The term ‘set of axioms’ is conceived broadly, to include whatever statements are admitted into a proof without demonstration, including axioms, postulates, and definitions.

The five Peano axioms are:

1.   Zero is a natural number.

2.   Every natural number has a successor in the natural numbers.

3.   Zero is not the successor of any natural number.

4.   If the successor of two natural numbers is the same, then the two original numbers are the same.

5.   If a set contains zero and the successor of every number is in the set, then the set contains the natural numbers.

Proof of the statement ‘1+1=2’ in the axiomatic system of Peano Arithmetic. For this proof, we need the definitions and axioms s0=1, s1=2, x+0=x, x+sy=s(x+y) from Peano Arithmetic. And the logical rules of inference P(r), r=t⇒P(t); P(v)⇒P(c) (where r, t; v; c; and P(t) range over terms; variables; constants; and propositions in the term t, respectively, and ‘⇒’ signifies logical implication. The following is a proof of 1+1=2: x+sy=s(x+y), 1+sy=s (1+y), 1+s0=s (1+0), x+0=x, 1+0=1, 1+s0=s1, s0=1, 1+1=s1, s1=2, 1+1=2

       An explanation of this proof is as follows. s0=1 [D1] and s1=2 [D2] are definitions of the constants 1 and 2, respectively, in Peano Arithmetic, x+0=x  [A1] and x+sy=s(x+y) [A2] are axioms of Peano Arithmetic. P(r), r=t⇒P(t) [R1] and P(v)⇒P(c) [R2], with the symbols as explained above, are logical rules of inference. The justification of the proof is below.

Proof of 1+1=2 with justification.

Step		Statement	Justification of Statement
S1		x+sy =s(x+y)	A2
S2		1+sy=s(1+y)	R2 applied to S1, using v=x, c=1
S3		1+s0 =s(1+0)	R2 applied to S2, using v=y, c=0
S4		X+0 =0	A1
S5		1+0 =1	R2 applied to S4, using v=x, c=1
S6		1+s0 =s1	R1 applied to S3 & S5, using r=1+0, t=1
S7		S0 =1	D1
S8		1+1 =s1	R1 applied to S6 & S7, using r=s0, t=1
S9		S1 =2	D2
S10	1+1 =2		R1 applied to s8 & S9, using r=s1, t=2

Note:

1.       The mathematical assumptions used are the definitions (D1 & D2).

2.       The axioms (A1 & A2).

3.       The logical assumptions are the rules of inference used (R1 & R2).