# Extension theorem (Existence)

1. May 19, 2010

### annoymage

1. The problem statement, all variables and given/known data

"Given a function H, assigning a value, 0 or 1, to each atomic sentence, define a sequence ℑ0, ℑ1, ℑ2, ℑ3,... of functions, as follows:
0 is just H.

Given a function ℑn, assigning a value, either 0 or 1, to the sentences of degree less than or equal to n, define the function ℑn+1, assigning a value, either 0 or 1, to the sentences of degree less than or equal to n+1, as follows: If φ has degree less than or equal to n, ℑn+1(φ) = ℑn(φ)."

I confused with the english, (sorry i'm not good in english). please clarify this

H({0,1}) = ℑ0, ℑ1, ℑ2, ℑ3,...
(does this mean "Given a function H, assigning a value, 0 or 1, to each atomic sentence, define a sequence ℑ0, ℑ1, ℑ2, ℑ3,... of functions"???)

and what "as follows: ℑ0 is just H." means? as follows? does is mean "such as"?

and i'll post the next question after this answered, because i wan to clear this first.

i'm sorry but the concise language is too concise for me i guess. help

2. May 19, 2010

### Susanne217

A thought experiment

Let x - 2 = 0 from this it follows that x = 2.

Understand the meaning now ?

Let try to formulate this another way let $$f(x) = x^2$$ which is defined as follows

$$- 1 \leq x \leq 1$$ that is the general meaning of the term "defined as follows" :)

3. May 19, 2010

### annoymage

so, anything after "as follows :" is the domain of the function?

which means,

H({0,1}) = {ℑ0, ℑ1, ℑ2, ℑ3,... }, and the domain is "ℑ0 is just H"

T_T i don't think it is correct of what i was doing

4. May 19, 2010

### Susanne217

For which subject is this for? Discrete mathematics ?

5. May 19, 2010

### annoymage

hmm, i'm doing self-study, and i guess the subject is "Mathematical Logic".

6. May 19, 2010

### Susanne217

Thats not my field but found some notes online in more plain english....

http://www.math.psu.edu/simpson/courses/math557/logic.pdf [Broken]