Implicit expression mathematics

The Paradox Elements of Jordan's Truth Value

See the set {x | x is a positive integer defined by a line of symbols} such as12317; The millionth prime number; The 23rd perfect number; And so on can define a positive integer. However, considering that the set {x | x is the smallest positive integer that one line of symbols can't define} contradicts the previous set, that is, one line of symbols can define but can't define positive integers.

Take mathematical induction as an example.

Prove it first.

1=(- 1)^0* 1( 1+ 1)/2= 1

Assume that the sum of n terms holds.

1-2^2+3^2-4^2+...+(- 1)^(n- 1)*n^2=(- 1)^(n- 1)*n*(n+ 1)/2

Sum of n+ 1

1-2^2+3^2-4^2+...+(- 1)^(n- 1)*n^2+(- 1)^(n)*(n+ 1)^2

=(- 1)^(n- 1)*n*(n+ 1)/2+(- 1)^(n)*(n+ 1)^2

=(- 1)^(n)*(n+ 1)^2-(- 1)^(n)*n*(n+ 1)/2

=(- 1)^(n)*[(n+ 1)^2-n*(n+ 1)/2]

=(- 1)^(n)*[(n+ 1)(n+ 1-n/2)]

=(- 1)^(n)*(n+ 1)(n+2)/2