alright well TI was one of my questions that I had so I'm glad you typed this out
-V_0 = the empty set which is transitive since y in V_0 is the empty set and the empty set is contained in the empty set
-V_a+1 I'm not sure, I know I have the power set in my notes I just can't find them right...
alrighty, now for V_alpha
V_alpha={a : rk(a)<alpha}
let rk(V_beta)= beta for all beta<alpha then V_beta is in V_alpha for every beta<alpha by the definition of V_alpha
do I just do the same thing as I did above pick a gamma and solve?
for some reason Latex is giving me grief
for a gamma in beta we have gamma<beta<alpha and thus gamma in beta in alpha
then gamma is in alpha meaning that beta is contained in alpha
is this a sound demonstration?
\gamma \in \beta
\gamma <\beta<\alpha
\gamma...
my definition is each ordinal a is the set of all smaller ordinals, i.e. a={B: B<a}
this mean B ε a
I'm not sure how to get the inclusion, I mean I know that B is included in a but is this obvious or should I show this?
Homework Statement
show every ordinal is a transitive set
show that every level V_a of the cumulative hierarchy is a transitive set
Homework Equations
The Attempt at a Solution
I understand that these are transitive sets, I'm just not sure how to show this. I feel like the...
Yea this was what I was trying to get at however you put it much more elegantly... My question before was how could I 'count' the number that come before m/n in a specific p. IE 3/1 is at position 5 we know p=2 + p=3 =3 but how can I include the fact that 1/3 and 2/2 come before 3/1 within the p=4
could I write that:
m+n=a_j and call this a set with a_j elements
then create a position set {{1/1},{1/2,2/1},{1/3,2/2,3/1},...} that is a_0,a_1,a_2 etc...
the position would be the addition of the number of elements in a_0+a_1+a_2...
I don't know where I'm going with this
Homework Statement
Prove that the fraction m/n occurs in position
\frac{m^2 +2mn + n^2 - m -3n}{2}
of the enumeration {1/1, 1/2, 2/1, 1/3, 2/2, 3/1,...}
of the set Q+ of positive rational numbers. (Hint: Count how many terms precede m/n in the enumeration.)
Homework Equations
The Attempt...
Homework Statement
describe exactly when
x intersecting (y union z) = (x intersecting y) union z
Homework Equations
The Attempt at a Solution
I just for some reason cannot see this solution and need a shove in the right direction