User blog comment:Googleaarex/My number is bigger! (Googology Wiki version)/@comment-2033667-20140919213150/@comment-25371339-20140921140206

My turn to troll.

Let [ ϕ ]      and [ ψ <span class="mo" id="MathJax-Span-55" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Main;">]       be Godel-coded formulas and <span aria-readonly="true" class="MathJax" id="MathJax-Element-9-Frame" role="textbox" style="display:inline;line-height:normal;word-spacing:normal;word-wrap:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;"><span class="math" id="MathJax-Span-56" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:static;width:0.598em;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:relative;width:0.436em;height:0px;font-size:18px;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;position:absolute;clip:rect(1.572em1000.003em2.33em-0.43em);top:-2.162em;left:0.003em;"><span class="mrow" id="MathJax-Span-57" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;"><span class="mi" id="MathJax-Span-58" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Math-italic;">s       and <span aria-readonly="true" class="MathJax" id="MathJax-Element-10-Frame" role="textbox" style="display:inline;line-height:normal;word-spacing:normal;word-wrap:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;"><span class="math" id="MathJax-Span-59" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:static;width:0.436em;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:relative;width:0.327em;height:0px;font-size:18px;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;position:absolute;clip:rect(1.356em1000.003em2.33em-0.484em);top:-2.162em;left:0.003em;"><span class="mrow" id="MathJax-Span-60" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;"><span class="mi" id="MathJax-Span-61" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Math-italic;">t       be variable assignments. Define <span aria-readonly="true" class="MathJax" id="MathJax-Element-11-Frame" role="textbox" style="display:inline;line-height:normal;word-spacing:normal;word-wrap:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;"><span class="math" id="MathJax-Span-62" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:static;width:5.522em;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:relative;width:4.169em;height:0px;font-size:18px;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;position:absolute;clip:rect(1.247em1000.003em2.6em-0.484em);top:-2.162em;left:0.003em;"><span class="mrow" id="MathJax-Span-63" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;"><span class="mtext" id="MathJax-Span-64" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Main;">Sat <span class="mo" id="MathJax-Span-65" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Main;">( <span class="mo" id="MathJax-Span-66" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Main;">[ <span class="mi" id="MathJax-Span-67" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Math-italic;">ϕ <span class="mo" id="MathJax-Span-68" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Main;">] <span class="mo" id="MathJax-Span-69" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Main;">, <span class="mi" id="MathJax-Span-70" style="padding-left:0.165em;vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Math-italic;">s <span class="mo" id="MathJax-Span-71" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Main;">)      as follows: ∀R { {    ∀[ψ], s: R([ψ],t) ↔ ([ψ] = "xᵢ ∈ xⱼ" ∧ t(xᵢ) ∈ t(xⱼ)) ∨ ([ψ] = "xᵢ = xⱼ" ∧ t(xᵢ) = t(xⱼ)) ∨ ([ψ] = "(¬θ)"   ∧ ¬R([θ], t)) ∨ ([ψ] = "([θ]∧ξ)" ∧ R([θ], t) ∧ R([ξ], t)) ∨ ([ψ] = "∃xᵢ(θ)" ∧ ∃t′: R([θ], t′)) (where t′ is a copy of t with xᵢ changed) } ⇒ R([ϕ],s) } Call a natural number <span aria-readonly="true" class="MathJax" id="MathJax-Element-12-Frame" role="textbox" style="display:inline;line-height:normal;word-spacing:normal;word-wrap:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;"><span class="math" id="MathJax-Span-72" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:static;width:1.139em;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:relative;width:0.869em;height:0px;font-size:18px;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;position:absolute;clip:rect(1.572em1000.003em2.33em-0.484em);top:-2.162em;left:0.003em;"><span class="mrow" id="MathJax-Span-73" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;"><span class="mi" id="MathJax-Span-74" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Math-italic;">m      "Rayo-namable in <span aria-readonly="true" class="MathJax" id="MathJax-Element-13-Frame" role="textbox" style="display:inline;line-height:normal;word-spacing:normal;word-wrap:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;"><span class="math" id="MathJax-Span-75" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:static;width:0.814em;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:relative;width:0.598em;height:0px;font-size:18px;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;position:absolute;clip:rect(1.572em1000.003em2.33em-0.484em);top:-2.162em;left:0.003em;"><span class="mrow" id="MathJax-Span-76" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;"><span class="mi" id="MathJax-Span-77" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Math-italic;">n       symbols" if there is a formula <span aria-readonly="true" class="MathJax" id="MathJax-Element-14-Frame" role="textbox" style="display:inline;line-height:normal;word-spacing:normal;word-wrap:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;"><span class="math" id="MathJax-Span-78" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:static;width:3.141em;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:relative;width:2.384em;height:0px;font-size:18px;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;position:absolute;clip:rect(1.247em1000.003em2.6em-0.43em);top:-2.162em;left:0.003em;"><span class="mrow" id="MathJax-Span-79" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;"><span class="mi" id="MathJax-Span-80" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Math-italic;">ϕ <span class="mo" id="MathJax-Span-81" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Main;">( <span class="msubsup" id="MathJax-Span-82" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:relative;width:0.977em;height:0px;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;position:absolute;clip:rect(1.572em1000.003em2.33em-0.43em);top:-2.162em;left:0.003em;"><span class="mi" id="MathJax-Span-83" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Math-italic;">x  <span style="vertical-align:0px;transition:none;-webkit-transition:none;position:absolute;top:-1.999em;left:0.544em;"><span class="mn" id="MathJax-Span-84" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-size:13px;font-family:MathJax_Main;">1    <span class="mo" id="MathJax-Span-85" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Main;">)       with less than <span aria-readonly="true" class="MathJax" id="MathJax-Element-15-Frame" role="textbox" style="display:inline;line-height:normal;word-spacing:normal;word-wrap:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;"><span class="math" id="MathJax-Span-86" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:static;width:0.814em;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:relative;width:0.598em;height:0px;font-size:18px;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;position:absolute;clip:rect(1.572em1000.003em2.33em-0.484em);top:-2.162em;left:0.003em;"><span class="mrow" id="MathJax-Span-87" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;"><span class="mi" id="MathJax-Span-88" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Math-italic;">n       symbols and <span aria-readonly="true" class="MathJax" id="MathJax-Element-16-Frame" role="textbox" style="display:inline;line-height:normal;word-spacing:normal;word-wrap:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;"><span class="math" id="MathJax-Span-89" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:static;width:1.356em;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:relative;width:1.031em;height:0px;font-size:18px;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;position:absolute;clip:rect(1.572em1000.003em2.492em-0.43em);top:-2.162em;left:0.003em;"><span class="mrow" id="MathJax-Span-90" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;"><span class="msubsup" id="MathJax-Span-91" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:relative;width:0.977em;height:0px;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;position:absolute;clip:rect(1.572em1000.003em2.33em-0.43em);top:-2.162em;left:0.003em;"><span class="mi" id="MathJax-Span-92" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Math-italic;">x  <span style="vertical-align:0px;transition:none;-webkit-transition:none;position:absolute;top:-1.999em;left:0.544em;"><span class="mn" id="MathJax-Span-93" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-size:13px;font-family:MathJax_Main;">1          as its only free variable that satisfies the following properties: <span aria-readonly="true" class="MathJax" id="MathJax-Element-24-Frame" role="textbox" style="display:inline;line-height:normal;word-spacing:normal;word-wrap:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;"><span class="math" id="MathJax-Span-147" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:static;width:4.71em;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:relative;width:3.574em;height:0px;font-size:18px;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;position:absolute;clip:rect(1.247em1000.003em2.6em-0.484em);top:-2.162em;left:0.003em;"><span class="mrow" id="MathJax-Span-148" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;"><span class="mtext" id="MathJax-Span-149" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Main;">Rayo <span class="mo" id="MathJax-Span-150" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Main;">( <span class="mi" id="MathJax-Span-151" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Math-italic;">n <span class="mo" id="MathJax-Span-152" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Main;">)    , then, is the smallest number greater than all numbers Rayo-namable in <span aria-readonly="true" class="MathJax" id="MathJax-Element-25-Frame" role="textbox" style="display:inline;line-height:normal;word-spacing:normal;word-wrap:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;"><span class="math" id="MathJax-Span-153" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:static;width:0.814em;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:relative;width:0.598em;height:0px;font-size:18px;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;position:absolute;clip:rect(1.572em1000.003em2.33em-0.484em);top:-2.162em;left:0.003em;"><span class="mrow" id="MathJax-Span-154" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;"><span class="mi" id="MathJax-Span-155" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Math-italic;">n       symbols.
 * 1) There is a variable assignment <span aria-readonly="true" class="MathJax" id="MathJax-Element-17-Frame" role="textbox" style="display:inline;line-height:normal;word-spacing:normal;word-wrap:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;"><span class="math" id="MathJax-Span-94" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:static;width:0.598em;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:relative;width:0.436em;height:0px;font-size:18px;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;position:absolute;clip:rect(1.572em1000.003em2.33em-0.43em);top:-2.162em;left:0.003em;"><span class="mrow" id="MathJax-Span-95" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;"><span class="mi" id="MathJax-Span-96" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Math-italic;">s    , assigning <span aria-readonly="true" class="MathJax" id="MathJax-Element-18-Frame" role="textbox" style="display:inline;line-height:normal;word-spacing:normal;word-wrap:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;"><span class="math" id="MathJax-Span-97" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:static;width:4.602em;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:relative;width:3.466em;height:0px;font-size:18px;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;position:absolute;clip:rect(1.572em1000.003em2.492em-0.43em);top:-2.162em;left:0.003em;"><span class="mrow" id="MathJax-Span-98" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;"><span class="msubsup" id="MathJax-Span-99" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:relative;width:0.977em;height:0px;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;position:absolute;clip:rect(1.572em1000.003em2.33em-0.43em);top:-2.162em;left:0.003em;"><span class="mi" id="MathJax-Span-100" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Math-italic;">x  <span style="vertical-align:0px;transition:none;-webkit-transition:none;position:absolute;top:-1.999em;left:0.544em;"><span class="mn" id="MathJax-Span-101" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-size:13px;font-family:MathJax_Main;">1    <span class="mo" id="MathJax-Span-102" style="padding-left:0.273em;vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Main;">: =  <span class="mi" id="MathJax-Span-103" style="padding-left:0.273em;vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Math-italic;">m      , such that <span aria-readonly="true" class="MathJax" id="MathJax-Element-19-Frame" role="textbox" style="display:inline;line-height:normal;word-spacing:normal;word-wrap:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;"><span class="math" id="MathJax-Span-104" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:static;width:7.903em;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:relative;width:5.955em;height:0px;font-size:18px;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;position:absolute;clip:rect(1.247em1000.003em2.6em-0.484em);top:-2.162em;left:0.003em;"><span class="mrow" id="MathJax-Span-105" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;"><span class="mtext" id="MathJax-Span-106" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Main;">Sat <span class="mo" id="MathJax-Span-107" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Main;">( <span class="mo" id="MathJax-Span-108" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Main;">[ <span class="mi" id="MathJax-Span-109" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Math-italic;">ϕ <span class="mo" id="MathJax-Span-110" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Main;">( <span class="msubsup" id="MathJax-Span-111" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:relative;width:0.977em;height:0px;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;position:absolute;clip:rect(1.572em1000.003em2.33em-0.43em);top:-2.162em;left:0.003em;"><span class="mi" id="MathJax-Span-112" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Math-italic;">x  <span style="vertical-align:0px;transition:none;-webkit-transition:none;position:absolute;top:-1.999em;left:0.544em;"><span class="mn" id="MathJax-Span-113" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-size:13px;font-family:MathJax_Main;">1    <span class="mo" id="MathJax-Span-114" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Main;">) <span class="mo" id="MathJax-Span-115" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Main;">] <span class="mo" id="MathJax-Span-116" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Main;">, <span class="mi" id="MathJax-Span-117" style="padding-left:0.165em;vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Math-italic;">s <span class="mo" id="MathJax-Span-118" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Main;">).
 * 2) For any variable assignment <span aria-readonly="true" class="MathJax" id="MathJax-Element-20-Frame" role="textbox" style="display:inline;line-height:normal;word-spacing:normal;word-wrap:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;"><span class="math" id="MathJax-Span-119" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:static;width:0.436em;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:relative;width:0.327em;height:0px;font-size:18px;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;position:absolute;clip:rect(1.356em1000.003em2.33em-0.484em);top:-2.162em;left:0.003em;"><span class="mrow" id="MathJax-Span-120" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;"><span class="mi" id="MathJax-Span-121" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Math-italic;">t    , if <span aria-readonly="true" class="MathJax" id="MathJax-Element-21-Frame" role="textbox" style="display:inline;line-height:normal;word-spacing:normal;word-wrap:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;"><span class="math" id="MathJax-Span-122" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:static;width:7.741em;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:relative;width:5.847em;height:0px;font-size:18px;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;position:absolute;clip:rect(1.247em1000.003em2.6em-0.484em);top:-2.162em;left:0.003em;"><span class="mrow" id="MathJax-Span-123" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;"><span class="mtext" id="MathJax-Span-124" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Main;">Sat <span class="mo" id="MathJax-Span-125" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Main;">( <span class="mo" id="MathJax-Span-126" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Main;">[ <span class="mi" id="MathJax-Span-127" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Math-italic;">ϕ <span class="mo" id="MathJax-Span-128" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Main;">( <span class="msubsup" id="MathJax-Span-129" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:relative;width:0.977em;height:0px;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;position:absolute;clip:rect(1.572em1000.003em2.33em-0.43em);top:-2.162em;left:0.003em;"><span class="mi" id="MathJax-Span-130" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Math-italic;">x  <span style="vertical-align:0px;transition:none;-webkit-transition:none;position:absolute;top:-1.999em;left:0.544em;"><span class="mn" id="MathJax-Span-131" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-size:13px;font-family:MathJax_Main;">1    <span class="mo" id="MathJax-Span-132" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Main;">) <span class="mo" id="MathJax-Span-133" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Main;">] <span class="mo" id="MathJax-Span-134" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Main;">, <span class="mi" id="MathJax-Span-135" style="padding-left:0.165em;vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Math-italic;">t <span class="mo" id="MathJax-Span-136" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Main;">)      , <span aria-readonly="true" class="MathJax" id="MathJax-Element-22-Frame" role="textbox" style="display:inline;line-height:normal;word-spacing:normal;word-wrap:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;"><span class="math" id="MathJax-Span-137" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:static;width:0.436em;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:relative;width:0.327em;height:0px;font-size:18px;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;position:absolute;clip:rect(1.356em1000.003em2.33em-0.484em);top:-2.162em;left:0.003em;"><span class="mrow" id="MathJax-Span-138" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;"><span class="mi" id="MathJax-Span-139" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Math-italic;">t       must have <span aria-readonly="true" class="MathJax" id="MathJax-Element-23-Frame" role="textbox" style="display:inline;line-height:normal;word-spacing:normal;word-wrap:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;"><span class="math" id="MathJax-Span-140" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:static;width:4.223em;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:relative;width:3.195em;height:0px;font-size:18px;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;position:absolute;clip:rect(1.572em1000.003em2.492em-0.43em);top:-2.162em;left:0.003em;"><span class="mrow" id="MathJax-Span-141" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;"><span class="msubsup" id="MathJax-Span-142" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline-block;position:relative;width:0.977em;height:0px;"><span style="vertical-align:0px;transition:none;-webkit-transition:none;position:absolute;clip:rect(1.572em1000.003em2.33em-0.43em);top:-2.162em;left:0.003em;"><span class="mi" id="MathJax-Span-143" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Math-italic;">x  <span style="vertical-align:0px;transition:none;-webkit-transition:none;position:absolute;top:-1.999em;left:0.544em;"><span class="mn" id="MathJax-Span-144" style="vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-size:13px;font-family:MathJax_Main;">1    <span class="mo" id="MathJax-Span-145" style="padding-left:0.273em;vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Main;">= <span class="mi" id="MathJax-Span-146" style="padding-left:0.273em;vertical-align:0px;transition:none;-webkit-transition:none;display:inline;position:static;font-family:MathJax_Math-italic;">m.

lol i copy it