The document discusses partitions of R3 (3-dimensional real space) into unit circles with and without the Axiom of Choice. It notes that without Choice, R3 can be partitioned into circles, but with Choice it can be partitioned into unit circles. The proof shows any partial partition into unit circles can be extended to a complete partition. However, sometimes there is not enough space to extend a partial partition. The document then discusses using Cohen forcing to construct a model where there is a partition of unit circles but no well-ordering of the real numbers. It presents lemmas on extending partial partitions and the transcendence degree when adding Cohen or other real numbers.

1. Partitions of R^3 in unit circles
and the Axiom of Choice
Set Theory in the UK 15.09.22
Azul Lihuen Fatalini
Universität Münster
Joint work with Prof. Ralf Schindler

2. Partitions of R^3 in unit circles
and the Axiom of Choice
Set Theory in the UK 15.09.22
Azul Lihuen Fatalini
Universität Münster
Joint work with Prof. Ralf Schindler

3. Coming back
I think it's about time that you guys find the right general
theorem for this (…). It seems like a source of a lot of
papers, and it seems to me that there is some bigger
theorem in the background that you're just scratching
instances of each and every time.

4. Context
Axiom
of choice
~
Paradoxical
sets

5. Context
Vitali set
Hamel basis
Axiom
Of Choice
My
Paradoxical
Mazurkiewicz set
sets
PUC
well - order
of the reals <
?
0 @ 0

6. Partitions in circles
PUC : =
partition of Pi in unit circles
Question : lit why Ñ ?
Iii)
Why partitions? ( 1-
covering
)
Ciii) Why circles ? Why unit circles?
( i) (ii
)
@
"
"
O

7. Partitions in circles
-
Theorem ( ZF) (Salkin)
1123 con be
porlihioned in circles .
Theorem (ZFC) (Conway
-
Croft / Kharazishvili)
1133 can be
partitioned in unit circles .

8. Partitions in circles
-
Theorem ( ZF) (Salkin)
1123 con be
partitioned in circles .
-
/
r
-
_
it
I 1 I
/
'
- -
'
- _
/

9. Partitions in unit circles
Theorem (ZFC) (Conway
-
Croft / Ktharazishvili)
1133 can be
partitioned in unit circles .
Let 1133 =
{ pin}* ± .
(O) (1) (2)
In "
€
.
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•
Bn
①
•
%
•
P
✗

10. Partitions in unit circles
Observation:
the proof shows thot
any portal PUC of
eordinolity-E.com be extended to a (complete) PUC .
Question : Com we
obuoys extend a
portal PUC to a
( complete) PUC ?

11. Partitions in unit circles
Observation:
the proof shows thot
any portal PUC of
cordiality =L can be extended to a (complete) PUC .
Question : Com we ◦
buoys extend a
portal PUC to a
( complete) PUC ?
-
Sometimes there is not enough
"
space
"
to extend a
portal PUC .

12. The result
Theorem
Zhou is a model of 2- Ft no well- order
of IR + F PUC

13. The model(s)
1 .
Cohen -
Halpern -
Lévy model :
H := HODLER
AU{A}
where
g
is ECW) -
generic
over L,
and
A =
{ Cn : new} is the set of Cohen reols added
by g.
2.
= [(IR,
b)
"%ʰ]
where
§ is ICCW,
) -
generic
over L ,
h is P -
generic
over
Lbj] ,
and
b = Uh is the PUC added
by h .

14. Cohen-Halpern-Lévy model
1 .
Cohen -
Halpern -
Lévy model :
H := HODLER
AU{A}
where
g
is ECW) -
generic
over L,
and
A =
{ Cn : new} is the set of Cohen reels added
by g.
Facts about H
( i) there is no well -
ordering of the reels .
Iii
) there is no countable subset of A.
kid IRNH =
a.eu#j.w4RnLEa])

15. Construction of a PUC in H
Q: How do we
get a PUC in H ?
U
Ra
RAH =
a¥jwlRnL[a]) =
¥w lawn
a ≤ A
IRNH ☒{a.bids
_ R{a.by/Rsra3lRga.cy
n =3 R∅
☒{by Rlc}
" "
§g"§g . '
-
'
' '
n=2
☒
{b.c
}
'
4441.
.
_
/ 111 n=^

16. Construction of a PUC in H
The
problems that arise
☒{a.b.c
}
☒{a.b.c
}
R{a
}
/R{any
R{a
}
☒{aib}
↑
☒{aib}
>
☒
{2,4
R∅ R∅
R{by Rlc} R{by Rlc}
☒
{b.c
}
☒
{b.c
}
Extendability (Strong) Amalgamation

17. Construction of a PUC in H
Lemma 1 ( Extendability)
Let V be a ZFC model and per such thot
VK
"
pis alportiol) PUC
"
.
Let c be a Cohen real over V.
Zhen,
there is EV[c] st Vcc] f-
"
≥p
^ is a PUC
"
Up = { 12}✗≤ [
"
✗
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→ EP
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o o o
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18. Construction of a PUC in H
Fact:
Let VEZFC and V[c
] be a
generic
extension obtained
by adding one Cohen red .
then the transcendence degree
of IRV
"]
over
IRV is E.

19. Construction of a PUC in H
{a.bids
R{a.by/Rsra3lRga.cy
R∅
>
R{by Rlc}
☒
{be}
(strong) Amalgamation @ =D

20. Construction of a PUC in H
Lemma 2 (strong Amalgamation)
Let a. b ,
c
mutually generic
Cohen reels and let p,
,
,
,
be such thot
[[a
] k p is a PUC
L [a.b] t
,
is a PUC
[[a. c] K
,
is a PUC
and
,
'
,
≤
p P.
Zhen Lla, b.c) t
,U ,
is a
portal PUC and it
can be extended to a PUC
≤pU,

21. Algebraic detour
Fact:
Let VFZFC and V[c
] be a
generic
extension obtained
by adding one Cohen red .
then the transcendence degree
of IRV
"]
over
IRV is E .
Lemma (Transcendence degree)
Let V be a model of ZFC and lets be a
finite set of
mutually generic
Cohen reels .
alg
then thetranscendence degree of
IÑ
"
]
over
.
U
[→
TES
11-1=151-1
is E .
Q : Wlwt can we
say if the reels are not Cohen reels ?

22. Coming back
Vitali set
Hamel basis
Axiom
Of Choice
My
Paradoxical
Mazurkiewicz set
sets
PUC
well - order
of the reals <
?
0 @ 0

23. Thank you for you attention!

24. References
[1] P. Bankston and R. Fox, Topological partitions of euclidean space by spheres, The American
Mathematical Monthly, 92 (1985), p. 423.
[2] M. Beriashvili and R. Schindler, Mazurkiewicz sets with no well-ordering of the reals,
Georgian Mathematical Journal. 2022;29(3): 343-345. https://doi.org/10.1515/gmj-2022-2142
[3] M. Beriashvili, R. Schindler, L. Wu, and L. Yu, Hamel bases and well-ordering the continuum,
in Proc. Amer. Math. Soc (Vol. 146, pp. 3565-3573), (2018).
[4] J. Brendle, F. Castiblanco, R. Schindler, L. Wu, and L. Yu, A model with everything except for
a well-ordering of the reals, (2018).
[5] J. H. Conway and H. T. Croft, Covering a sphere with congruent great-circle arcs,
Mathematical Proceedings of the Cambridge Philosophical Society, 60 (1964), pp. 787–800.
[6] M. Jonsson and J. Wästlund, Partitions of ℝ 3 into curves, Mathematica Scandinavica, 83.
10.7146/math.scand.a-13850 , (1998).
[7] A. B. Kharazishvili, Partition of the three-dimensional space into congruent circles, Bull.
Acad. Sci. GSSR, v. 119, n. 1, 1985 (in Russian).
[8] A. B. Kharazishvili and T. S. Tetunashvili, On some coverings of the euclidean plane with
pairwise congruent circles, American Mathematical Monthly, 117 (2010), pp. 414–423.
[9] J. Schilhan, Maximal sets without choice, (2022).
[10] A. Szulkin, R3 is the union of disjoint circles, (1983).
[11] Z. Vidnyánszky, Transfinite inductions producing coanalytic sets, Fundamenta
Mathematicae. 224. 10.4064/fm224-2-2 , (2012)