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Kanamori.json
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Kanamori.json
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"fields": [
"measurable",
"there is a \\(\\kappa \\) complete ultrafilter on \\(\\kappa \\)",
"there is a normal ultrafilter on \\(\\kappa \\)",
"there is a \\(j:V\\prec M\\) with \\(crit(j)=\\kappa\\)",
"if \\(U\\) is a normal ultrafilter on \\(\\kappa\\) then for every \\(f:[\\kappa ]^{<\\omega}\\rightarrow \\gamma\\) where \\(\\gamma<\\kappa\\) there is a homogeneous set for \\(f\\) in \\(U\\)",
"measurable cardinals are \\(\\Pi_1^2\\)-indescribable. Moreover, for every \\(\\Pi_1^2\\) formula \\(\\varphi\\), the set of \\(\\alpha\\) which reflect \\(\\varphi\\) is in every normal filter.<div>Also, \\(\\{\\alpha <\\kappa |\\alpha \\)is totally indescribable\\(\\}\\in U\\) for every normal filter on \\(\\kappa\\).</div>"
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"\\(\\kappa \\) is regular and strong limit",
"",
"",
"if \\(\\kappa \\rightarrow (\\kappa )^2_2 \\) and \\(\\kappa > \\omega \\)then \\(\\kappa \\) is inaccesible",
"\\(\\kappa\\) is inaccessible iff it is \\(\\Sigma^1_1\\)-indescribable"
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"\\(\\{\\alpha <\\kappa | \\alpha\\) is inaccessible \\(\\}\\) is stationary in \\(\\kappa\\)",
"",
"",
"",
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"note_model_uuid": "3c0803c2-09fb-11e9-a4a0-801934c598f6",
"tags": [
"cardinals"
]
},
{
"__type__": "Note",
"data": "",
"fields": [
"weakly compact",
"any collection of \\(L_{\\kappa\\kappa}\\) sentences using at most \\(\\kappa\\) non-logical symbols if \\(\\kappa\\)-satisfiable, is satisfiable",
"",
"\\(\\kappa\\) has the extension property: for every \\(R\\subseteq V_\\kappa\\) there is a transitive \\(X\\neq V_\\kappa\\) and \\(S\\subseteq X\\) such that \\(\\langle V_\\kappa,\\in,R\\rangle\\prec\\langle X,\\in,S\\rangle\\)",
"\\(\\kappa\\) is weakly compact iff \\(\\kappa\\rightarrow (\\kappa)^n_\\lambda \\) for every \\(n<\\omega ,\\lambda<\\kappa\\) and iff \\(\\kappa\\rightarrow (\\kappa)^2_2\\)",
"\\(\\kappa\\) is weakly compact iff it is \\(\\Pi^1_1\\)-indescribable. Also, if \\(\\kappa\\) is weakly compact then \\(\\kappa\\) is weakly compact in \\(L\\)"
],
"flags": 0,
"guid": "hzqE8>6+[]",
"note_model_uuid": "3c0803c2-09fb-11e9-a4a0-801934c598f6",
"tags": [
"cardinals"
]
},
{
"__type__": "Note",
"data": "",
"fields": [
"strongly compact",
"any \\(\\kappa\\)- satisfiable collection of \\(L_{\\kappa\\kappa}\\) sentences is satisfiable",
"Every \\(\\kappa\\)-complete filter can be extended to a \\(\\kappa\\)-complete ultrafilter",
"",
"",
""
],
"flags": 0,
"guid": "MQ{PWjsP$*",
"note_model_uuid": "3c0803c2-09fb-11e9-a4a0-801934c598f6",
"tags": [
"cardinals"
]
},
{
"__type__": "Note",
"data": "",
"fields": [
"\\(\\gamma\\)-supercompact (\\(\\kappa\\) is supercompact if it is \\(\\gamma\\)-supercompact for every \\(\\gamma\\))",
"there is a \\(j:V\\prec M\\) with \\(crit(j)=\\kappa\\), \\(\\gamma <j(\\kappa )\\) and such that \\(^\\gamma M\\subseteq M\\)",
"For \\(\\gamma \\geq\\kappa\\), \\(\\kappa\\) is \\(\\gamma\\)-supercompact iff there is a normal ultrafilter over \\(\\mathcal{P}_\\kappa \\gamma \\)",
"there is a \\(j:V\\prec M\\) with \\(crit(j)=\\kappa\\), \\(\\gamma <j(\\kappa )\\) and such that \\(^\\gamma M\\subseteq M\\)",
"",
"If \\(\\kappa\\) is supercompact, then \\(V_\\kappa\\prec_2 V\\)<div>\\(\\kappa\\) is supercompact iff for every \\(\\eta >\\kappa\\) there is an \\(\\alpha <\\kappa\\) and \\(e:V_\\alpha\\prec V_\\eta\\) with \\(e(crit(e))=\\kappa\\)</div>"
],
"flags": 0,
"guid": "Er):@tBOD$",
"note_model_uuid": "3c0803c2-09fb-11e9-a4a0-801934c598f6",
"tags": [
"cardinals"
]
},
{
"__type__": "Note",
"data": "",
"fields": [
"\\(\\gamma\\)-compact (\\(\\kappa\\) is strongly compact iff it is \\(\\gamma\\)-compact for every \\(\\gamma\\geq\\kappa\\))",
"There is a fine ultrafilter over \\(\\mathcal{P}_\\kappa \\gamma\\)",
"Every \\(\\kappa\\)-complete filter generated by at most \\(|\\gamma |\\) sets can be extended to a \\(\\kappa\\)-complete ultrafilter",
"There is a \\(j:V\\prec M\\) with \\(crit(j)=\\kappa\\) such that for any \\(X\\subseteq M\\) with \\(|X|\\leq \\gamma\\) there is a \\(Y\\in M\\) such that \\(Y\\supseteq X\\) and \\(M\\models |Y|<j(\\kappa )\\)",
"",
""
],
"flags": 0,
"guid": "noXW&i*u=7",
"note_model_uuid": "3c0803c2-09fb-11e9-a4a0-801934c598f6",
"tags": [
"cardinals"
]
},
{
"__type__": "Note",
"data": "",
"fields": [
"Define when \\(\\kappa\\) is \\(Q\\)-indescribable for \\(Q\\) a set of sentences",
"if for any \\(R\\subseteq V_\\kappa\\) and \\(Q\\) sentence \\(\\varphi\\) such that \\(\\langle V_\\kappa ,\\in ,R\\rangle \\models\\varphi\\) there is an \\(\\alpha <\\kappa\\) such that \\(\\langle V_\\alpha ,\\in ,R\\cap V_\\alpha\\rangle\\models\\varphi\\)"
],
"flags": 0,
"guid": "gQu~<5>)]f",
"note_model_uuid": "3c0803d7-09fb-11e9-a4a0-801934c598f6",
"tags": [
"cardinals"
]
},
{
"__type__": "Note",
"data": "",
"fields": [
"Quote Kunen's inconsistency theorem",
"If \\(j:V\\prec M\\) then \\(M\\neq V\\)"
],
"flags": 0,
"guid": "p+=p&/?wUJ",
"note_model_uuid": "3c0803d7-09fb-11e9-a4a0-801934c598f6",
"tags": []
},
{
"__type__": "Note",
"data": "",
"fields": [
"Define when \\(\\beta\\rightarrow(\\alpha )_\\delta^\\gamma \\)",
"When for any \\(f:[\\beta ]^\\gamma\\rightarrow\\delta\\) there is an \\(H\\in [\\beta ]^\\alpha\\) homogeneous for \\(f\\): \\(|f^{\\prime\\prime}[H]^\\gamma |\\leq 1\\)"
],
"flags": 0,
"guid": "Lz3CCKaxEq",
"note_model_uuid": "3c0803d7-09fb-11e9-a4a0-801934c598f6",
"tags": [
"partitions"
]
},
{
"__type__": "Note",
"data": "",
"fields": [
"\\(\\kappa\\) is Ramsey",
"iff \\(\\kappa\\rightarrow(\\kappa)^{<\\omega}_2"
],
"flags": 0,
"guid": "PoNDlUBvmI",
"note_model_uuid": "3c0803e0-09fb-11e9-a4a0-801934c598f6",
"tags": [
"partitions"
]
},
{
"__type__": "Note",
"data": "",
"fields": [
"Define Chang's conjecture \\(\\langle\\kappa,\\lambda\\rangle\\twoheadrightarrow\\langle\\mu,\\nu\\rangle\\)",
"Whenever \\(\\mathcal{A}\\) is a structure of type \\(\\langle\\kappa,\\lambda\\rangle\\) there is a \\(\\mathcal{B}\\prec\\mathcal{A}\\) of type \\(\\langle\\mu,\\nu\\rangle\\)"
],
"flags": 0,
"guid": "cX*ekC<|VM",
"note_model_uuid": "3c0803d7-09fb-11e9-a4a0-801934c598f6",
"tags": [
"partitions"
]
},
{
"__type__": "Note",
"data": "",
"fields": [
"Define when \\(\\beta\\rightarrow[\\alpha]^\\gamma_\\delta\\)",
"If for any \\(f:[\\beta]^\\gamma\\rightarrow\\delta\\) there is an \\(H\\in[\\beta]^\\alpha\\) such that \\(f^{\\prime\\prime}[H]^\\gamma\\neq\\delta\\)"
],
"flags": 0,
"guid": "HFm7U$~mNI",
"note_model_uuid": "3c0803d7-09fb-11e9-a4a0-801934c598f6",
"tags": [
"partitions"
]
},
{
"__type__": "Note",
"data": "",
"fields": [
"What is the connection between Chang's conjecture and partition relations?",
"\\(\\langle\\kappa,\\lambda\\rangle\\twoheadrightarrow\\langle\\mu,<\\nu\\rangle\\) iff \\(\\kappa\\rightarrow[\\mu]^{<\\omega}_{\\lambda,<\\nu}\\)"
],
"flags": 0,
"guid": "B0A|XQk?2A",
"note_model_uuid": "3c0803d7-09fb-11e9-a4a0-801934c598f6",
"tags": [
"partitions"
]
},
{
"__type__": "Note",
"data": "",
"fields": [
"Define when \\(\\beta\\rightarrow[\\alpha]^\\gamma_{\\delta,<\\eta}\\)",
"The same definition, with the requirement that \\(|f^{\\prime\\prime}[H]^\\gamma|<\\eta"
],
"flags": 0,
"guid": "dBv-+cYk~h",
"note_model_uuid": "3c0803d7-09fb-11e9-a4a0-801934c598f6",
"tags": [
"partitions"
]
},
{
"__type__": "Note",
"data": "",
"fields": [
"When is \\(kappa\\) \\(\\nu\\)-Rowbottom?",
"if \\(\\kappa\\rightarrow[\\kappa]^{<\\omega}_{\\lambda, <\\nu}\\) for every \\(\\lambda<\\kappa\\)"
],
"flags": 0,
"guid": "HawF.$yml&",
"note_model_uuid": "3c0803d7-09fb-11e9-a4a0-801934c598f6",
"tags": [
"cardinals",
"partitions"
]
},
{
"__type__": "Note",
"data": "",
"fields": [
"When is \\(\\kappa\\) a Rowbottom cardinal?",
"If it is \\(\\omega_1\\)-Rowbottom. Namely, if if \\(\\kappa\\rightarrow[\\kappa]^{<\\omega}_{\\lambda, <\\omega_1}\\) for every \\(\\lambda<\\kappa\\)"
],
"flags": 0,
"guid": "DuW+uy(CWd",
"note_model_uuid": "3c0803d7-09fb-11e9-a4a0-801934c598f6",
"tags": [
"cardinals",
"partitions"
]
},
{
"__type__": "Note",
"data": "",
"fields": [
"When is \\(\\kappa\\) a Jonsson cardinal?",
"If every algebra of cardinality \\(\\kappa\\) has a proper subalgebra of the same cardinality"
],
"flags": 0,
"guid": "KN9)4INUt:",
"note_model_uuid": "3c0803f5-09fb-11e9-a4a0-801934c598f6",
"tags": [
"cardinals",
"partitions"
]
},
{
"__type__": "Note",
"data": "",
"fields": [
"\\(\\kappa\\) being a Jonsson cardinal is equivalent to the partition relation:",
"\\(\\kappa\\rightarrow[\\kappa]^{<\\omega}_\\kappa\\) is equivalent to \\(\\kappa\\) being"
],
"flags": 0,
"guid": "n|u]rW=*mi",
"note_model_uuid": "3c0803f8-09fb-11e9-a4a0-801934c598f6",
"tags": [
"partitions"
]
},
{
"__type__": "Note",
"data": "",
"fields": [
"What can be said about the least Jonsson cardinal?",
"It is either weakly inaccessible or singular of cofinality \\(\\omega\\)<div>Also, it is \\(\\nu\\)-Rowbottom for some \\(\\nu\\) (and \\(\\nu\\) can be taken to be the least \\(\\delta\\) such that \\(\\kappa\\rightarrow[\\kappa]^{<\\omega}_\\delta\\))</div>"
],
"flags": 0,
"guid": "r.!Vpo}x9L",
"note_model_uuid": "3c0803f5-09fb-11e9-a4a0-801934c598f6",
"tags": [
"partitions"
]
},
{
"__type__": "Note",
"data": "",
"fields": [
"What conditions are enough for \\(\\kappa^+\\) not to be a Jonsson cardinal?",
"If \\(\\kappa\\) is not Jonsson<div>If \\(2^\\kappa=\\kappa^+\\)</div><div>If \\kappa is regular</div><div>or if \\(\\kappa\\) is singular and not the limit of regular Jonsson cardinals</div>"
],
"flags": 0,
"guid": "L7zl!%SG;a",
"note_model_uuid": "3c0803f5-09fb-11e9-a4a0-801934c598f6",
"tags": [
"partitions"
]
},
{
"__type__": "Note",
"data": "",
"fields": [
"Define when \\(\\langle X,<\\rangle\\) is a set of indiscernibles for \\(\\mathcal{M}\\)",
"if for every formula \\(\\varphi(v_1,\\cdots,v_n)\\) in the language of \\mathcal{M} and for every \\(x_1<\\cdots<x_n\\) and \\(y_1<\\cdots<y_n\\) all in \\(X\\), \\(\\mathcal{M}\\models\\varphi[x_1,\\cdots,x_n]\\iff \\mathcal{M}\\models\\varphi[y_1,\\cdots,y_n]\\)"
],
"flags": 0,
"guid": "xB5!~ku>_G",
"note_model_uuid": "3c0803f5-09fb-11e9-a4a0-801934c598f6",
"tags": [
"sharps"
]
},
{
"__type__": "Note",
"data": "",
"fields": [
"What is an EM blueprint?",
"It is the theory in \\(\\mathcal{L}^*_\\in\\) of some structure \\(\\langle L_\\delta,\\in,x_k\\rangle_{k\\in\\omega}\\) where \\(\\delta>\\omega\\) is limit ordinal and \\(\\{x_k|k\\in\\omega\\}\\) is a set of ordinal indiscernibles for \\(\\langle L_\\delta,\\in\\rangle\\) indexed in increasing order"
],
"flags": 0,
"guid": "glLSi4[Ftv",
"note_model_uuid": "3c0803f5-09fb-11e9-a4a0-801934c598f6",
"tags": [
"sharps"
]
},
{
"__type__": "Note",
"data": "",
"fields": [
"What is property \\((I)\\) of \\(T=0^#\\)?",
"\\(\\mathcal{M}(T,\\alpha)\\) is well founded for every \\(\\alpha<\\omega_1\\) (and it is equivalent for \\(\\mathcal{M}(T,\\alpha)\\) being well founded for every \\(\\alpha\\))"
],
"flags": 0,
"guid": "s.@i?;BA$}",
"note_model_uuid": "3c0803f5-09fb-11e9-a4a0-801934c598f6",
"tags": [
"sharps"
]
},
{
"__type__": "Note",
"data": "",
"fields": [
"What is property \\((II)\\) of \\(T=0^#\\)?",
"For any \\(n\\)-ary Skolem term \\(t\\), \\(T\\) contains the sentence \\(t(c_0,\\cdots,c_{n-1})\\in On\\rightarrow t(c_0,\\cdots,c_{n-1})<c_n\\)."
],
"flags": 0,
"guid": "oYH]N9b*v(",
"note_model_uuid": "3c0803f5-09fb-11e9-a4a0-801934c598f6",
"tags": [
"sharps"
]
},
{
"__type__": "Note",
"data": "",
"fields": [
"What is property \\((III)\\) of \\(T=0^#\\)?",
"For any \\(n+m+1\\)-ary Skolem term \\(t\\), \\(T\\) contains the sentence \\(t(c_0,\\cdots,c_{m+n})<c_m\\rightarrow t(c_0,\\cdots,c_{m+n})=t(c_0,\\cdots,c_{m-1},c_{m+n+1},\\cdots,c_{m+2n+1})\\)."
],
"flags": 0,
"guid": "KZr&g{^s_C",
"note_model_uuid": "3c0803f5-09fb-11e9-a4a0-801934c598f6",
"tags": [
"sharps"
]
},
{
"__type__": "Note",
"data": "",
"fields": [
"What is the name of the first singular cardinal refuting GCH?",
"Steve."
],
"flags": 0,
"guid": "KBQn:*2i_<",
"note_model_uuid": "3c0803f5-09fb-11e9-a4a0-801934c598f6",
"tags": []
}
]
}